On Log-Concave Functions

Andrea Colesanti Universit`adi Firenze

On log-concave functions

Analytic Tools in Probability and Applications IMA – University of Minnesota

April 27 - May 1, 2015 I Log-concave functions and their addition

I The Pr´ekopa-Leindler inequality and its inﬁnitesimal form

I Functional forms of Blaschke-Santal´oand Rogers-Shephard inequalities

I Valuations on log-concave functions

Outline of the talk I The Pr´ekopa-Leindler inequality and its inﬁnitesimal form

I Functional forms of Blaschke-Santal´oand Rogers-Shephard inequalities

I Valuations on log-concave functions

Outline of the talk

I Log-concave functions and their addition I Functional forms of Blaschke-Santal´oand Rogers-Shephard inequalities

I Valuations on log-concave functions

Outline of the talk

I Log-concave functions and their addition

I The Pr´ekopa-Leindler inequality and its inﬁnitesimal form I Valuations on log-concave functions

Outline of the talk

I Log-concave functions and their addition

I The Pr´ekopa-Leindler inequality and its inﬁnitesimal form

I Functional forms of Blaschke-Santal´oand Rogers-Shephard inequalities Outline of the talk

I Log-concave functions and their addition

I The Pr´ekopa-Leindler inequality and its inﬁnitesimal form

I Functional forms of Blaschke-Santal´oand Rogers-Shephard inequalities

I Valuations on log-concave functions A function n f : R → R+ is said to be log-concave if

f = e−u

where

n −∞ u : R → R ∪ {+∞} is convex (e = 0).

We will always assume

lim u(x) = +∞ ⇒ lim f (x) = 0, |x|→+∞ |x|→+∞

and set

n n C = {f log-concave on R : lim|x|→+∞ f (x) = 0}.

Log-concave functions (e−∞ = 0).

We will always assume

lim u(x) = +∞ ⇒ lim f (x) = 0, |x|→+∞ |x|→+∞

and set

n n C = {f log-concave on R : lim|x|→+∞ f (x) = 0}.

Log-concave functions A function n f : R → R+ is said to be log-concave if

f = e−u

where

n u : R → R ∪ {+∞} is convex We will always assume

lim u(x) = +∞ ⇒ lim f (x) = 0, |x|→+∞ |x|→+∞

and set

n n C = {f log-concave on R : lim|x|→+∞ f (x) = 0}.

Log-concave functions A function n f : R → R+ is said to be log-concave if

f = e−u

where

n −∞ u : R → R ∪ {+∞} is convex (e = 0). and set

n n C = {f log-concave on R : lim|x|→+∞ f (x) = 0}.

Log-concave functions A function n f : R → R+ is said to be log-concave if

f = e−u

where

n −∞ u : R → R ∪ {+∞} is convex (e = 0).

We will always assume

lim u(x) = +∞ ⇒ lim f (x) = 0, |x|→+∞ |x|→+∞ Log-concave functions A function n f : R → R+ is said to be log-concave if

f = e−u

where

n −∞ u : R → R ∪ {+∞} is convex (e = 0).

We will always assume

lim u(x) = +∞ ⇒ lim f (x) = 0, |x|→+∞ |x|→+∞

and set

n n C = {f log-concave on R : lim|x|→+∞ f (x) = 0}. I The Gaussian function:

2 f (x) = e−kxk /2.

I The characteristic function of a convex body K (compact n convex subset of R ): 1 if x ∈ K, f (x) = χ (x) = K 0 if x ∈/ K.

Note that f = e−IK where 0 if x ∈ K I (x) = = indicatrix function of K. K +∞ if x ∈/ K

Examples I The characteristic function of a convex body K (compact n convex subset of R ): 1 if x ∈ K, f (x) = χ (x) = K 0 if x ∈/ K.

Note that f = e−IK where 0 if x ∈ K I (x) = = indicatrix function of K. K +∞ if x ∈/ K

Examples

I The Gaussian function:

2 f (x) = e−kxk /2. Note that f = e−IK where 0 if x ∈ K I (x) = = indicatrix function of K. K +∞ if x ∈/ K

Examples

I The Gaussian function:

2 f (x) = e−kxk /2.

I The characteristic function of a convex body K (compact n convex subset of R ): 1 if x ∈ K, f (x) = χ (x) = K 0 if x ∈/ K. = indicatrix function of K.

Examples

I The Gaussian function:

2 f (x) = e−kxk /2.

I The characteristic function of a convex body K (compact n convex subset of R ): 1 if x ∈ K, f (x) = χ (x) = K 0 if x ∈/ K.

Note that f = e−IK where 0 if x ∈ K I (x) = K +∞ if x ∈/ K Examples

I The Gaussian function:

2 f (x) = e−kxk /2.

I The characteristic function of a convex body K (compact n convex subset of R ): 1 if x ∈ K, f (x) = χ (x) = K 0 if x ∈/ K.

Note that f = e−IK where 0 if x ∈ K I (x) = = indicatrix function of K. K +∞ if x ∈/ K Log-concave functions are natural objects to study, in analysis and in probability.

I Probability measures with a log-concave density. Isoperimetry; spectral gap (or Poincar´e)inequalities; log-Sobolev inequalities; concentration phenomena; central limit theorems...

I Geometrization of analysis. Study log-concave functions in parallel with the theory of convex bodies (convex geometry). Functional forms of classical geometric inequalities in convex geometry; duality; functional versions of mixed and intrinsic volumes; valuations... We will focus on the second aspect.

Motivations I Probability measures with a log-concave density. Isoperimetry; spectral gap (or Poincar´e)inequalities; log-Sobolev inequalities; concentration phenomena; central limit theorems...

Motivations

Log-concave functions are natural objects to study, in analysis and in probability. I Geometrization of analysis. Study log-concave functions in parallel with the theory of convex bodies (convex geometry). Functional forms of classical geometric inequalities in convex geometry; duality; functional versions of mixed and intrinsic volumes; valuations... We will focus on the second aspect.

Motivations

Log-concave functions are natural objects to study, in analysis and in probability.

I Probability measures with a log-concave density. Isoperimetry; spectral gap (or Poincar´e)inequalities; log-Sobolev inequalities; concentration phenomena; central limit theorems... We will focus on the second aspect.

Motivations

Log-concave functions are natural objects to study, in analysis and in probability.

I Probability measures with a log-concave density. Isoperimetry; spectral gap (or Poincar´e)inequalities; log-Sobolev inequalities; concentration phenomena; central limit theorems...

Log-concave functions are natural objects to study, in analysis and in probability.

I Probability measures with a log-concave density. Isoperimetry; spectral gap (or Poincar´e)inequalities; log-Sobolev inequalities; concentration phenomena; central limit theorems...

(f ⊕ g)(z) = sup{f (x)g(y) | x + y = z} .

Consistently, for t > 0 we deﬁne z (t · f )(z) = f t t (so that, e.g., f ⊕ f = 2 · f ). Cn is closed with respect to these operations.

Addition of log-concave functions Consistently, for t > 0 we deﬁne z (t · f )(z) = f t t (so that, e.g., f ⊕ f = 2 · f ). Cn is closed with respect to these operations.

Addition of log-concave functions

For f , g ∈ Cn we set

(f ⊕ g)(z) = sup{f (x)g(y) | x + y = z} . (so that, e.g., f ⊕ f = 2 · f ). Cn is closed with respect to these operations.

Addition of log-concave functions

For f , g ∈ Cn we set

(f ⊕ g)(z) = sup{f (x)g(y) | x + y = z} .

Consistently, for t > 0 we deﬁne z (t · f )(z) = f t t Cn is closed with respect to these operations.

Addition of log-concave functions

For f , g ∈ Cn we set

(f ⊕ g)(z) = sup{f (x)g(y) | x + y = z} .

Consistently, for t > 0 we deﬁne z (t · f )(z) = f t t (so that, e.g., f ⊕ f = 2 · f ). Addition of log-concave functions

For f , g ∈ Cn we set

(f ⊕ g)(z) = sup{f (x)g(y) | x + y = z} .

Consistently, for t > 0 we deﬁne z (t · f )(z) = f t t (so that, e.g., f ⊕ f = 2 · f ). Cn is closed with respect to these operations. n Given K and L convex bodies in R and α, β ≥ 0

αK + βL = {αx + βy : x ∈ K, y ∈ L}.

We have the following relation

χαK+βL = α · χK ⊕ β · χL.

In other words, ⊕ extends the Minkowski addition at the level of log-concave functions.

Minkowski addition and ⊕

The ﬁrst motivation for the deﬁnition of ⊕ is its connection to the Minkowski addition of convex bodies. We have the following relation

χαK+βL = α · χK ⊕ β · χL.

In other words, ⊕ extends the Minkowski addition at the level of log-concave functions.

Minkowski addition and ⊕

The ﬁrst motivation for the deﬁnition of ⊕ is its connection to the Minkowski addition of convex bodies.

n Given K and L convex bodies in R and α, β ≥ 0

αK + βL = {αx + βy : x ∈ K, y ∈ L}. In other words, ⊕ extends the Minkowski addition at the level of log-concave functions.

Minkowski addition and ⊕

The ﬁrst motivation for the deﬁnition of ⊕ is its connection to the Minkowski addition of convex bodies.

n Given K and L convex bodies in R and α, β ≥ 0

αK + βL = {αx + βy : x ∈ K, y ∈ L}.

We have the following relation

χαK+βL = α · χK ⊕ β · χL. Minkowski addition and ⊕

The ﬁrst motivation for the deﬁnition of ⊕ is its connection to the Minkowski addition of convex bodies.

n Given K and L convex bodies in R and α, β ≥ 0

αK + βL = {αx + βy : x ∈ K, y ∈ L}.

We have the following relation

χαK+βL = α · χK ⊕ β · χL.

In other words, ⊕ extends the Minkowski addition at the level of log-concave functions. (f ⊕ g)(z) = sup{f (x)g(y) | x + y = z} If we write f = e−u and g = e−v then f ⊕ g = e−w where

w(z) = inf{u(x) + v(y) | x + y = z} = inf-convolution of u and v = uv (extensively studied in Rockafellar’s monograph Convex analysis). Note that

epi(uv) = epi(u) + epi(v).

More on ⊕ If we write f = e−u and g = e−v then f ⊕ g = e−w where

w(z) = inf{u(x) + v(y) | x + y = z} = inf-convolution of u and v = uv (extensively studied in Rockafellar’s monograph Convex analysis). Note that

epi(uv) = epi(u) + epi(v).

More on ⊕

(f ⊕ g)(z) = sup{f (x)g(y) | x + y = z} If we write f = e−u and g = e−v then f ⊕ g = e−w where

w(z) = inf{u(x) + v(y) | x + y = z} = inf-convolution of u and v = uv (extensively studied in Rockafellar’s monograph Convex analysis). More on ⊕

(f ⊕ g)(z) = sup{f (x)g(y) | x + y = z} If we write f = e−u and g = e−v then f ⊕ g = e−w where

w(z) = inf{u(x) + v(y) | x + y = z} = inf-convolution of u and v = uv (extensively studied in Rockafellar’s monograph Convex analysis). Note that

epi(uv) = epi(u) + epi(v). Another property of inf-convolution:

∗ ∗ ∗ (uv) = u + v where ∗ is the usual conjugate of convex functions:

u∗(y) = sup [(x, y) − u(x)]. x∈Rn

Summarizing,

∗ ∗ ∗ f ⊕ g = e−u ⊕ e−v = e−(uv) = e−(u +v ) ,

i.e. ⊕ coincides with the usual addition applied to the conjugate of the exponent (with sign changed) of log-concave functions.

The role of the conjugate function Summarizing,

∗ ∗ ∗ f ⊕ g = e−u ⊕ e−v = e−(uv) = e−(u +v ) ,

i.e. ⊕ coincides with the usual addition applied to the conjugate of the exponent (with sign changed) of log-concave functions.

The role of the conjugate function

Another property of inf-convolution:

∗ ∗ ∗ (uv) = u + v where ∗ is the usual conjugate of convex functions:

u∗(y) = sup [(x, y) − u(x)]. x∈Rn i.e. ⊕ coincides with the usual addition applied to the conjugate of the exponent (with sign changed) of log-concave functions.

The role of the conjugate function

Another property of inf-convolution:

∗ ∗ ∗ (uv) = u + v where ∗ is the usual conjugate of convex functions:

u∗(y) = sup [(x, y) − u(x)]. x∈Rn

Summarizing,

∗ ∗ ∗ f ⊕ g = e−u ⊕ e−v = e−(uv) = e−(u +v ) , The role of the conjugate function

Another property of inf-convolution:

∗ ∗ ∗ (uv) = u + v where ∗ is the usual conjugate of convex functions:

u∗(y) = sup [(x, y) − u(x)]. x∈Rn

Summarizing,

∗ ∗ ∗ f ⊕ g = e−u ⊕ e−v = e−(uv) = e−(u +v ) ,

i.e. ⊕ coincides with the usual addition applied to the conjugate of the exponent (with sign changed) of log-concave functions. The total mass of a log-concave function f is simply its integral: Z I (f ) = fdx. Rn It is not diﬃcult to see that our assumptions imply

0 ≤ I (f ) < ∞ ∀ f .

I (f ) is the counterpart of the volume (Lebesgue measure) of a convex body K, in convex geometry, denoted by Vn:

Vn(K) = volume of K.

The total mass of a log-concave function It is not diﬃcult to see that our assumptions imply

0 ≤ I (f ) < ∞ ∀ f .

I (f ) is the counterpart of the volume (Lebesgue measure) of a convex body K, in convex geometry, denoted by Vn:

Vn(K) = volume of K.

The total mass of a log-concave function

The total mass of a log-concave function f is simply its integral: Z I (f ) = fdx. Rn I (f ) is the counterpart of the volume (Lebesgue measure) of a convex body K, in convex geometry, denoted by Vn:

Vn(K) = volume of K.

The total mass of a log-concave function

The total mass of a log-concave function f is simply its integral: Z I (f ) = fdx. Rn It is not diﬃcult to see that our assumptions imply

0 ≤ I (f ) < ∞ ∀ f . The total mass of a log-concave function

The total mass of a log-concave function f is simply its integral: Z I (f ) = fdx. Rn It is not diﬃcult to see that our assumptions imply

0 ≤ I (f ) < ∞ ∀ f .

I (f ) is the counterpart of the volume (Lebesgue measure) of a convex body K, in convex geometry, denoted by Vn:

Vn(K) = volume of K. For every f , g log-concave and for every t ∈ [0, 1]

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn (Pr´ekopa, 1971; Leindler 1972). Equality holds (up to some degenerate cases) iﬀ

n g(y) = f (α(y − y0)), ∀ y ∈ R ,

n for some y0 ∈ R and α > 0 (Dubuc, 1979).

f → log(I (f )) is concave.

The Pr´ekopa-Leindler inequality Equality holds (up to some degenerate cases) iﬀ

n g(y) = f (α(y − y0)), ∀ y ∈ R ,

n for some y0 ∈ R and α > 0 (Dubuc, 1979).

f → log(I (f )) is concave.

The Pr´ekopa-Leindler inequality

For every f , g log-concave and for every t ∈ [0, 1]

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn (Pr´ekopa, 1971; Leindler 1972). f → log(I (f )) is concave.

The Pr´ekopa-Leindler inequality

For every f , g log-concave and for every t ∈ [0, 1]

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn (Pr´ekopa, 1971; Leindler 1972). Equality holds (up to some degenerate cases) iﬀ

n g(y) = f (α(y − y0)), ∀ y ∈ R ,

n for some y0 ∈ R and α > 0 (Dubuc, 1979). The Pr´ekopa-Leindler inequality

For every f , g log-concave and for every t ∈ [0, 1]

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn (Pr´ekopa, 1971; Leindler 1972). Equality holds (up to some degenerate cases) iﬀ

n g(y) = f (α(y − y0)), ∀ y ∈ R ,

n for some y0 ∈ R and α > 0 (Dubuc, 1979).

f → log(I (f )) is concave. Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn

For f = χK and g = χL (K and L convex bodies) we get:

1−t t Vn((1 − t)K + tL) ≥ Vn(K) Vn(L) ,

and then, by a simple argument based on homogeneity of volume,

1/n 1/n 1/n Vn((1 − t)K + tL) ≥ (1 − t)Vn(K) + tVn(L) . (BM)

(BM) is among the most important results in convex geometry. It is connected to the isoperimetric inequality and to the family of Aleksandrov-Fenchel inequalities. See the exhausitve survey by Gardner: The Brunn-Minkowski inequality, 2002.

Pr´ekopa-Leindler and Brunn-Minkowski inequality For f = χK and g = χL (K and L convex bodies) we get:

1−t t Vn((1 − t)K + tL) ≥ Vn(K) Vn(L) ,

and then, by a simple argument based on homogeneity of volume,

1/n 1/n 1/n Vn((1 − t)K + tL) ≥ (1 − t)Vn(K) + tVn(L) . (BM)

Pr´ekopa-Leindler and Brunn-Minkowski inequality

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn and then, by a simple argument based on homogeneity of volume,

1/n 1/n 1/n Vn((1 − t)K + tL) ≥ (1 − t)Vn(K) + tVn(L) . (BM)

Pr´ekopa-Leindler and Brunn-Minkowski inequality

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn

For f = χK and g = χL (K and L convex bodies) we get:

1−t t Vn((1 − t)K + tL) ≥ Vn(K) Vn(L) , (BM) is among the most important results in convex geometry. It is connected to the isoperimetric inequality and to the family of Aleksandrov-Fenchel inequalities. See the exhausitve survey by Gardner: The Brunn-Minkowski inequality, 2002.

Pr´ekopa-Leindler and Brunn-Minkowski inequality

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn

For f = χK and g = χL (K and L convex bodies) we get:

1−t t Vn((1 − t)K + tL) ≥ Vn(K) Vn(L) ,

and then, by a simple argument based on homogeneity of volume,

1/n 1/n 1/n Vn((1 − t)K + tL) ≥ (1 − t)Vn(K) + tVn(L) . (BM) See the exhausitve survey by Gardner: The Brunn-Minkowski inequality, 2002.

Pr´ekopa-Leindler and Brunn-Minkowski inequality

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn

For f = χK and g = χL (K and L convex bodies) we get:

1−t t Vn((1 − t)K + tL) ≥ Vn(K) Vn(L) ,

and then, by a simple argument based on homogeneity of volume,

1/n 1/n 1/n Vn((1 − t)K + tL) ≥ (1 − t)Vn(K) + tVn(L) . (BM)

(BM) is among the most important results in convex geometry. It is connected to the isoperimetric inequality and to the family of Aleksandrov-Fenchel inequalities. Pr´ekopa-Leindler and Brunn-Minkowski inequality

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn

For f = χK and g = χL (K and L convex bodies) we get:

1−t t Vn((1 − t)K + tL) ≥ Vn(K) Vn(L) ,

and then, by a simple argument based on homogeneity of volume,

1/n 1/n 1/n Vn((1 − t)K + tL) ≥ (1 − t)Vn(K) + tVn(L) . (BM)

I It can be viewed as a reverse form of the H¨olderinequality, recalling that

(1 − t) · f ⊕ t · g(z) = sup f 1−t (x)g t (y). (1−t)x+ty=z

I It is a special case of a family of inequalities proved by Barthe, which reverse the Borell-Brascamp-Lieb inequalities.

I It is also a limit case of the reverse Young inequalities for convolutions.

More remarks about (PL) I It can be viewed as a reverse form of the H¨olderinequality, recalling that

(1 − t) · f ⊕ t · g(z) = sup f 1−t (x)g t (y). (1−t)x+ty=z

I It is a special case of a family of inequalities proved by Barthe, which reverse the Borell-Brascamp-Lieb inequalities.

I It is also a limit case of the reverse Young inequalities for convolutions.

More remarks about (PL)

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn I It is a special case of a family of inequalities proved by Barthe, which reverse the Borell-Brascamp-Lieb inequalities.

I It is also a limit case of the reverse Young inequalities for convolutions.

More remarks about (PL)

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn

I It can be viewed as a reverse form of the H¨olderinequality, recalling that

(1 − t) · f ⊕ t · g(z) = sup f 1−t (x)g t (y). (1−t)x+ty=z I It is also a limit case of the reverse Young inequalities for convolutions.

More remarks about (PL)

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn

I It can be viewed as a reverse form of the H¨olderinequality, recalling that

(1 − t) · f ⊕ t · g(z) = sup f 1−t (x)g t (y). (1−t)x+ty=z

I It is a special case of a family of inequalities proved by Barthe, which reverse the Borell-Brascamp-Lieb inequalities. More remarks about (PL)

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn

I It can be viewed as a reverse form of the H¨olderinequality, recalling that

(1 − t) · f ⊕ t · g(z) = sup f 1−t (x)g t (y). (1−t)x+ty=z

I It is a special case of a family of inequalities proved by Barthe, which reverse the Borell-Brascamp-Lieb inequalities.

I It is also a limit case of the reverse Young inequalities for convolutions. Bobkov and Ledoux (2000) showed how to obtain a Poincar´etype inequality due to Brascamp and Lieb (see next slide), and the log-Sobolev inequality starting from the (PL) inequality. In 2007 they also provided a proof of the Sobolev inequality via the Brunn-Minkowski inequality.

In general it is now well understood that concavity inequalities like (PL) and (BM) can be used to prove integral inequalities involving derivatives.

More ambitiously, we would like to see that (PL) is in fact equivalent to a class of integral inequalities of Poincar´etype (precisely those of Brascamp and Lieb).

From (PL) to Poincar´eand Sobolev inequalities In general it is now well understood that concavity inequalities like (PL) and (BM) can be used to prove integral inequalities involving derivatives.

More ambitiously, we would like to see that (PL) is in fact equivalent to a class of integral inequalities of Poincar´etype (precisely those of Brascamp and Lieb).

From (PL) to Poincar´eand Sobolev inequalities

Bobkov and Ledoux (2000) showed how to obtain a Poincar´etype inequality due to Brascamp and Lieb (see next slide), and the log-Sobolev inequality starting from the (PL) inequality. In 2007 they also provided a proof of the Sobolev inequality via the Brunn-Minkowski inequality. More ambitiously, we would like to see that (PL) is in fact equivalent to a class of integral inequalities of Poincar´etype (precisely those of Brascamp and Lieb).

From (PL) to Poincar´eand Sobolev inequalities

In general it is now well understood that concavity inequalities like (PL) and (BM) can be used to prove integral inequalities involving derivatives. From (PL) to Poincar´eand Sobolev inequalities

In general it is now well understood that concavity inequalities like (PL) and (BM) can be used to prove integral inequalities involving derivatives.

More ambitiously, we would like to see that (PL) is in fact equivalent to a class of integral inequalities of Poincar´etype (precisely those of Brascamp and Lieb). n Thm. (Brascamp-Lieb, ’76). Let µ be a probability measure on R with density f . Assume that

−u 2 n 2 n f = e , u ∈ C (R ), D u(x) > 0 ∀ x ∈ R .

1 n Then for every φ ∈ C (R ), Z Z Z (φ − φdµ)2dµ ≤ ((D2u)−1∇φ, ∇φ)dµ. (BL) Rn Rn Rn

Remark. This result applies to the Gaussian measure γn and provides the standard Poincar´einequality for this measure.

A Poincar´einequality for log-concave measures by Brascamp and Lieb 2 n D u(x) > 0 ∀ x ∈ R .

1 n Then for every φ ∈ C (R ), Z Z Z (φ − φdµ)2dµ ≤ ((D2u)−1∇φ, ∇φ)dµ. (BL) Rn Rn Rn

Remark. This result applies to the Gaussian measure γn and provides the standard Poincar´einequality for this measure.

A Poincar´einequality for log-concave measures by Brascamp and Lieb

n Thm. (Brascamp-Lieb, ’76). Let µ be a probability measure on R with density f . Assume that

−u 2 n f = e , u ∈ C (R ), 1 n Then for every φ ∈ C (R ), Z Z Z (φ − φdµ)2dµ ≤ ((D2u)−1∇φ, ∇φ)dµ. (BL) Rn Rn Rn

Remark. This result applies to the Gaussian measure γn and provides the standard Poincar´einequality for this measure.

A Poincar´einequality for log-concave measures by Brascamp and Lieb

n Thm. (Brascamp-Lieb, ’76). Let µ be a probability measure on R with density f . Assume that

−u 2 n 2 n f = e , u ∈ C (R ), D u(x) > 0 ∀ x ∈ R . Remark. This result applies to the Gaussian measure γn and provides the standard Poincar´einequality for this measure.

A Poincar´einequality for log-concave measures by Brascamp and Lieb

n Thm. (Brascamp-Lieb, ’76). Let µ be a probability measure on R with density f . Assume that

−u 2 n 2 n f = e , u ∈ C (R ), D u(x) > 0 ∀ x ∈ R .

1 n Then for every φ ∈ C (R ), Z Z Z (φ − φdµ)2dµ ≤ ((D2u)−1∇φ, ∇φ)dµ. (BL) Rn Rn Rn A Poincar´einequality for log-concave measures by Brascamp and Lieb

n Thm. (Brascamp-Lieb, ’76). Let µ be a probability measure on R with density f . Assume that

−u 2 n 2 n f = e , u ∈ C (R ), D u(x) > 0 ∀ x ∈ R .

1 n Then for every φ ∈ C (R ), Z Z Z (φ − φdµ)2dµ ≤ ((D2u)−1∇φ, ∇φ)dµ. (BL) Rn Rn Rn

Remark. This result applies to the Gaussian measure γn and provides the standard Poincar´einequality for this measure. Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn This is equivalent to say that Z n J : C → R, J(f ) = log fdx Rn is concave; hence J00(f ) ≤ 0, ∀f . Claim:

J00(f ) ≤ 0 ⇔ (BL) for the measure µ: dµ = f

(whenever f = e−u, D2u > 0).

The inﬁnitesimal form of (PL) (PL)

This is equivalent to say that Z n J : C → R, J(f ) = log fdx Rn is concave; hence J00(f ) ≤ 0, ∀f . Claim:

J00(f ) ≤ 0 ⇔ (BL) for the measure µ: dµ = f

(whenever f = e−u, D2u > 0).

The inﬁnitesimal form of (PL)

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy Rn Rn Rn hence J00(f ) ≤ 0, ∀f . Claim:

J00(f ) ≤ 0 ⇔ (BL) for the measure µ: dµ = f

(whenever f = e−u, D2u > 0).

The inﬁnitesimal form of (PL)

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn This is equivalent to say that Z n J : C → R, J(f ) = log fdx Rn is concave; Claim:

J00(f ) ≤ 0 ⇔ (BL) for the measure µ: dµ = f

(whenever f = e−u, D2u > 0).

The inﬁnitesimal form of (PL)

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn This is equivalent to say that Z n J : C → R, J(f ) = log fdx Rn is concave; hence J00(f ) ≤ 0, ∀f . (whenever f = e−u, D2u > 0).

The inﬁnitesimal form of (PL)

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn This is equivalent to say that Z n J : C → R, J(f ) = log fdx Rn is concave; hence J00(f ) ≤ 0, ∀f . Claim:

J00(f ) ≤ 0 ⇔ (BL) for the measure µ: dµ = f The inﬁnitesimal form of (PL)

Z Z 1−t Z t [(1 − t) · f ⊕ t · g]dz ≥ fdx gdy (PL) Rn Rn Rn This is equivalent to say that Z n J : C → R, J(f ) = log fdx Rn is concave; hence J00(f ) ≤ 0, ∀f . Claim:

J00(f ) ≤ 0 ⇔ (BL) for the measure µ: dµ = f

(whenever f = e−u, D2u > 0). J00(f ) can be thought as a bilinear quadratic form acting on test functions h, deﬁned as follows

2 00 d (J (f )h, h) = 2 J(f ⊕ t · h) . dt t=0

The condition (J00(f )h, h) ≤ 0 ∀h turns out to be exactly the Poincar´einequality of Brascamp and Lieb.

A formal computation of J00(f ) The condition (J00(f )h, h) ≤ 0 ∀h turns out to be exactly the Poincar´einequality of Brascamp and Lieb.

A formal computation of J00(f )

J00(f ) can be thought as a bilinear quadratic form acting on test functions h, deﬁned as follows

2 00 d (J (f )h, h) = 2 J(f ⊕ t · h) . dt t=0 A formal computation of J00(f )

J00(f ) can be thought as a bilinear quadratic form acting on test functions h, deﬁned as follows

2 00 d (J (f )h, h) = 2 J(f ⊕ t · h) . dt t=0

The condition (J00(f )h, h) ≤ 0 ∀h turns out to be exactly the Poincar´einequality of Brascamp and Lieb. −u 2 n 2 Let f = e with u ∈ C (R ) and D u(x) > 0 for every x. Then u∗ has the same properties of u and

∇u = (∇u∗)−1, D2u(x) = [D2u∗(∇u(x))]−1.

Moreover u(x) = x · ∇u(x) − u∗(∇u(x)).

Z Z I (f ) = fdx = e−udx (y = ∇u(x)) Rn Rn Z ∗ ∗ = eu (y)−y·∇u (y) det(D2u∗(y))dy. Rn

In practice: write I (f ) in terms of u∗ Then u∗ has the same properties of u and

∇u = (∇u∗)−1, D2u(x) = [D2u∗(∇u(x))]−1.

Moreover u(x) = x · ∇u(x) − u∗(∇u(x)).

Z Z I (f ) = fdx = e−udx (y = ∇u(x)) Rn Rn Z ∗ ∗ = eu (y)−y·∇u (y) det(D2u∗(y))dy. Rn

In practice: write I (f ) in terms of u∗

−u 2 n 2 Let f = e with u ∈ C (R ) and D u(x) > 0 for every x. Moreover u(x) = x · ∇u(x) − u∗(∇u(x)).

Z Z I (f ) = fdx = e−udx (y = ∇u(x)) Rn Rn Z ∗ ∗ = eu (y)−y·∇u (y) det(D2u∗(y))dy. Rn

In practice: write I (f ) in terms of u∗

−u 2 n 2 Let f = e with u ∈ C (R ) and D u(x) > 0 for every x. Then u∗ has the same properties of u and

∇u = (∇u∗)−1, D2u(x) = [D2u∗(∇u(x))]−1. Z Z I (f ) = fdx = e−udx (y = ∇u(x)) Rn Rn Z ∗ ∗ = eu (y)−y·∇u (y) det(D2u∗(y))dy. Rn

In practice: write I (f ) in terms of u∗

−u 2 n 2 Let f = e with u ∈ C (R ) and D u(x) > 0 for every x. Then u∗ has the same properties of u and

∇u = (∇u∗)−1, D2u(x) = [D2u∗(∇u(x))]−1.

Moreover u(x) = x · ∇u(x) − u∗(∇u(x)). (y = ∇u(x))

Z ∗ ∗ = eu (y)−y·∇u (y) det(D2u∗(y))dy. Rn

In practice: write I (f ) in terms of u∗

−u 2 n 2 Let f = e with u ∈ C (R ) and D u(x) > 0 for every x. Then u∗ has the same properties of u and

∇u = (∇u∗)−1, D2u(x) = [D2u∗(∇u(x))]−1.

Moreover u(x) = x · ∇u(x) − u∗(∇u(x)).

Z Z I (f ) = fdx = e−udx Rn Rn Z ∗ ∗ = eu (y)−y·∇u (y) det(D2u∗(y))dy. Rn

In practice: write I (f ) in terms of u∗

−u 2 n 2 Let f = e with u ∈ C (R ) and D u(x) > 0 for every x. Then u∗ has the same properties of u and

∇u = (∇u∗)−1, D2u(x) = [D2u∗(∇u(x))]−1.

Moreover u(x) = x · ∇u(x) − u∗(∇u(x)).

Z Z I (f ) = fdx = e−udx (y = ∇u(x)) Rn Rn In practice: write I (f ) in terms of u∗

−u 2 n 2 Let f = e with u ∈ C (R ) and D u(x) > 0 for every x. Then u∗ has the same properties of u and

∇u = (∇u∗)−1, D2u(x) = [D2u∗(∇u(x))]−1.

Moreover u(x) = x · ∇u(x) − u∗(∇u(x)).

Z Z I (f ) = fdx = e−udx (y = ∇u(x)) Rn Rn Z ∗ ∗ = eu (y)−y·∇u (y) det(D2u∗(y))dy. Rn As ⊕ and · coincide with usual addition and multiplication on u∗, (PL) is now equivalent to:

u∗ → log(G(u∗)) concave.

Then 2 d ∗ 2 log(G(u + tψ)) ≤ 0 dt t=0 for an arbitrary test function ψ. This condition plus the reverse change of variable y = ∇u(x) lead to the inequality by Brascamp and Lieb.

Z Z ∗ ∗ I (f ) = fdx = eu (y)−y·∇u (y) det(D2u∗(y))dy =: G(u∗). Rn Rn Then 2 d ∗ 2 log(G(u + tψ)) ≤ 0 dt t=0 for an arbitrary test function ψ. This condition plus the reverse change of variable y = ∇u(x) lead to the inequality by Brascamp and Lieb.

Z Z ∗ ∗ I (f ) = fdx = eu (y)−y·∇u (y) det(D2u∗(y))dy =: G(u∗). Rn Rn As ⊕ and · coincide with usual addition and multiplication on u∗, (PL) is now equivalent to:

u∗ → log(G(u∗)) concave. ≤ 0

for an arbitrary test function ψ. This condition plus the reverse change of variable y = ∇u(x) lead to the inequality by Brascamp and Lieb.

Z Z ∗ ∗ I (f ) = fdx = eu (y)−y·∇u (y) det(D2u∗(y))dy =: G(u∗). Rn Rn As ⊕ and · coincide with usual addition and multiplication on u∗, (PL) is now equivalent to:

u∗ → log(G(u∗)) concave.

Then 2 d ∗ 2 log(G(u + tψ)) dt t=0 This condition plus the reverse change of variable y = ∇u(x) lead to the inequality by Brascamp and Lieb.

Z Z ∗ ∗ I (f ) = fdx = eu (y)−y·∇u (y) det(D2u∗(y))dy =: G(u∗). Rn Rn As ⊕ and · coincide with usual addition and multiplication on u∗, (PL) is now equivalent to:

u∗ → log(G(u∗)) concave.

Then 2 d ∗ 2 log(G(u + tψ)) ≤ 0 dt t=0 for an arbitrary test function ψ. Z Z ∗ ∗ I (f ) = fdx = eu (y)−y·∇u (y) det(D2u∗(y))dy =: G(u∗). Rn Rn As ⊕ and · coincide with usual addition and multiplication on u∗, (PL) is now equivalent to:

u∗ → log(G(u∗)) concave.

Then 2 d ∗ 2 log(G(u + tψ)) ≤ 0 dt t=0 for an arbitrary test function ψ. This condition plus the reverse change of variable y = ∇u(x) lead to the inequality by Brascamp and Lieb. I Kolesnikov and Milman (’14) extended these inequalities to a certain class of Riemannian manifolds and used their result to establish a (BM) inequality in this setting.

I Many inequalities of (BM) type (proved or conjectured) admit an infinitesimal version, typically consisting of a family of functional inequalities of Poincarétype, but in general it is difficult to use them to prove (BM). At this regard see the recent result by Livshyts, Marsiglietti, Nayar and Zvavitch about the conjectured dimensional version of (BM) in Gauss space, for symmetric convex bodies.

Remarks

I A similar procedure shows that the Brunn-Minkowski inequality is equivalent to a family of weighted Poincaré inequalities on the unit sphere ([C. ’08]). I Many inequalities of (BM) type (proved or conjectured) admit an infinitesimal version, typically consisting of a family of functional inequalities of Poincarétype, but in general it is difficult to use them to prove (BM). At this regard see the recent result by Livshyts, Marsiglietti, Nayar and Zvavitch about the conjectured dimensional version of (BM) in Gauss space, for symmetric convex bodies.

Remarks

I A similar procedure shows that the Brunn-Minkowski inequality is equivalent to a family of weighted Poincar´e inequalities on the unit sphere ([C. ’08]).

I Kolesnikov and Milman (’14) extended these inequalities to a certain class of Riemannian manifolds and used their result to establish a (BM) inequality in this setting. At this regard see the recent result by Livshyts, Marsiglietti, Nayar and Zvavitch about the conjectured dimensional version of (BM) in Gauss space, for symmetric convex bodies.

Remarks

I A similar procedure shows that the Brunn-Minkowski inequality is equivalent to a family of weighted Poincar´e inequalities on the unit sphere ([C. ’08]).

I Kolesnikov and Milman (’14) extended these inequalities to a certain class of Riemannian manifolds and used their result to establish a (BM) inequality in this setting.

I A similar procedure shows that the Brunn-Minkowski inequality is equivalent to a family of weighted Poincar´e inequalities on the unit sphere ([C. ’08]).

I Kolesnikov and Milman (’14) extended these inequalities to a certain class of Riemannian manifolds and used their result to establish a (BM) inequality in this setting.

Z Z ∗ e−udx · e−u dx ≤ (2π)n. Rn Rn Equality holds if and only if f is a Gaussian function.

Thm. (Blaschke-Santal´oinequality). Let K be a centrally symmetric convex body, and let K ◦ be its polar:

◦ n K = {y ∈ R : x · y ≤ 1 ∀ x ∈ K}.

Then ◦ 2 Vn(K) Vn(K ) ≤ Vn(Bn) ,

where Bn is the unit ball. Equality holds iﬀ K is an ellipsoid.

The functional version of Blaschke-Santal´oinequality Equality holds if and only if f is a Gaussian function.

Thm. (Blaschke-Santal´oinequality). Let K be a centrally symmetric convex body, and let K ◦ be its polar:

◦ n K = {y ∈ R : x · y ≤ 1 ∀ x ∈ K}.

Then ◦ 2 Vn(K) Vn(K ) ≤ Vn(Bn) ,

where Bn is the unit ball. Equality holds iﬀ K is an ellipsoid.

The functional version of Blaschke-Santal´oinequality

Thm. Let f = e−u be a log-concave function and assume that f is even. Then

Z Z ∗ e−udx · e−u dx ≤ (2π)n. Rn Rn Thm. (Blaschke-Santal´oinequality). Let K be a centrally symmetric convex body, and let K ◦ be its polar:

◦ n K = {y ∈ R : x · y ≤ 1 ∀ x ∈ K}.

Then ◦ 2 Vn(K) Vn(K ) ≤ Vn(Bn) ,

where Bn is the unit ball. Equality holds iﬀ K is an ellipsoid.

The functional version of Blaschke-Santal´oinequality

Thm. Let f = e−u be a log-concave function and assume that f is even. Then

Z Z ∗ e−udx · e−u dx ≤ (2π)n. Rn Rn Equality holds if and only if f is a Gaussian function. Then ◦ 2 Vn(K) Vn(K ) ≤ Vn(Bn) ,

where Bn is the unit ball. Equality holds iﬀ K is an ellipsoid.

The functional version of Blaschke-Santal´oinequality

Thm. Let f = e−u be a log-concave function and assume that f is even. Then

Z Z ∗ e−udx · e−u dx ≤ (2π)n. Rn Rn Equality holds if and only if f is a Gaussian function.

Thm. (Blaschke-Santal´oinequality). Let K be a centrally symmetric convex body, and let K ◦ be its polar:

◦ n K = {y ∈ R : x · y ≤ 1 ∀ x ∈ K}. The functional version of Blaschke-Santal´oinequality

Thm. Let f = e−u be a log-concave function and assume that f is even. Then

Z Z ∗ e−udx · e−u dx ≤ (2π)n. Rn Rn Equality holds if and only if f is a Gaussian function.

Thm. (Blaschke-Santal´oinequality). Let K be a centrally symmetric convex body, and let K ◦ be its polar:

◦ n K = {y ∈ R : x · y ≤ 1 ∀ x ∈ K}.

Then ◦ 2 Vn(K) Vn(K ) ≤ Vn(Bn) ,

where Bn is the unit ball. Equality holds iﬀ K is an ellipsoid. I The proof of the functional version of (BL) is due to Ball. The result was extended to the general (non-even) case by Artstein, Klartag and Milman. Further contributions were given by Fradelizi-Gordon-Reisner, Meyer, Lehec, Rotem.

I For both theorems, to ﬁnd an optimal lower bound for the corresponding products (of integrals or of volumes) is an open problem, for n ≥ 3.

Remarks I For both theorems, to ﬁnd an optimal lower bound for the corresponding products (of integrals or of volumes) is an open problem, for n ≥ 3.

Remarks

I The proof of the functional version of (BL) is due to Ball. The result was extended to the general (non-even) case by Artstein, Klartag and Milman. Further contributions were given by Fradelizi-Gordon-Reisner, Meyer, Lehec, Rotem. Remarks

I For both theorems, to ﬁnd an optimal lower bound for the corresponding products (of integrals or of volumes) is an open problem, for n ≥ 3. n Given a convex body K in R , its diﬀerence body DK is

DK = K + (−K) = {x − y : x, y ∈ K}.

Note that DK is centrally symmetric. By the Brunn-Minkowski inequality n Vn(DK) ≥ 2 Vn(K). Thm. (Rogers-Shephard inequality).

2n V (DK) ≤ V (K). n n n

Equality holds iﬀ K is a symplex.

The diﬀerence body inequality By the Brunn-Minkowski inequality n Vn(DK) ≥ 2 Vn(K). Thm. (Rogers-Shephard inequality).

2n V (DK) ≤ V (K). n n n

Equality holds iﬀ K is a symplex.

The diﬀerence body inequality

n Given a convex body K in R , its diﬀerence body DK is

DK = K + (−K) = {x − y : x, y ∈ K}.

Note that DK is centrally symmetric. Thm. (Rogers-Shephard inequality).

2n V (DK) ≤ V (K). n n n

Equality holds iﬀ K is a symplex.

The diﬀerence body inequality

n Given a convex body K in R , its diﬀerence body DK is

DK = K + (−K) = {x − y : x, y ∈ K}.

Note that DK is centrally symmetric. By the Brunn-Minkowski inequality n Vn(DK) ≥ 2 Vn(K). The diﬀerence body inequality

n Given a convex body K in R , its diﬀerence body DK is

DK = K + (−K) = {x − y : x, y ∈ K}.

Note that DK is centrally symmetric. By the Brunn-Minkowski inequality n Vn(DK) ≥ 2 Vn(K). Thm. (Rogers-Shephard inequality).

2n V (DK) ≤ V (K). n n n

Equality holds iﬀ K is a symplex. Let f be a log-concave function and set f¯(z) = f (−z), for every n z ∈ R . Deﬁne 1 1 Df (z) = · f ⊕ f¯ (z) = sup pf (x)f (−y). 2 2 x+y=2z

This is an even function. By the (PL) inequality: Z Z Df dz ≥ f dx. Rn Rn

A functional version Deﬁne 1 1 Df (z) = · f ⊕ f¯ (z) = sup pf (x)f (−y). 2 2 x+y=2z

This is an even function. By the (PL) inequality: Z Z Df dz ≥ f dx. Rn Rn

A functional version

Let f be a log-concave function and set f¯(z) = f (−z), for every n z ∈ R . = sup pf (x)f (−y). x+y=2z

This is an even function. By the (PL) inequality: Z Z Df dz ≥ f dx. Rn Rn

A functional version

Let f be a log-concave function and set f¯(z) = f (−z), for every n z ∈ R . Deﬁne 1 1 Df (z) = · f ⊕ f¯ (z) 2 2 By the (PL) inequality: Z Z Df dz ≥ f dx. Rn Rn

A functional version

Let f be a log-concave function and set f¯(z) = f (−z), for every n z ∈ R . Deﬁne 1 1 Df (z) = · f ⊕ f¯ (z) = sup pf (x)f (−y). 2 2 x+y=2z

This is an even function. A functional version

Let f be a log-concave function and set f¯(z) = f (−z), for every n z ∈ R . Deﬁne 1 1 Df (z) = · f ⊕ f¯ (z) = sup pf (x)f (−y). 2 2 x+y=2z

This is an even function. By the (PL) inequality: Z Z Df dz ≥ f dx. Rn Rn Equality holds iﬀ (up to linear transformations)

e−(x1+···+xn) if x ≥ 0 for every i, f (x ,..., x ) = i 1 n 0 elsewhere.

Extensions and other functional versions of the Rogers-Shephard inequality have been recently found by Artstein-Avidan, Einhorn, Florentin, Ostrover (2015), and Alonso-Gutiérrez,González, Jiménez,Villa (2015).

Thm. (C. ’06). Z Z Df dz ≤ 2n f dx. Rn Rn Extensions and other functional versions of the Rogers-Shephard inequality have been recently found by Artstein-Avidan, Einhorn, Florentin, Ostrover (2015), and Alonso-Gutiérrez,González, Jiménez,Villa (2015).

Thm. (C. ’06). Z Z Df dz ≤ 2n f dx. Rn Rn Equality holds iﬀ (up to linear transformations)

e−(x1+···+xn) if x ≥ 0 for every i, f (x ,..., x ) = i 1 n 0 elsewhere. Thm. (C. ’06). Z Z Df dz ≤ 2n f dx. Rn Rn Equality holds iﬀ (up to linear transformations)

e−(x1+···+xn) if x ≥ 0 for every i, f (x ,..., x ) = i 1 n 0 elsewhere.

µ(K ∪ L) + µ(K ∩ L) = µ(K) + µ(L)

for every K and L s.t. K ∪ L ∈ Kn. The study of valuations, especially with continuity and invariance properties, is one of the most active and proliﬁc branches of convex geometry, after the seminal contributions by Hadwiger, McMullen and Alesker.

Thm. (Volume theorem). Let µ be a rigid motion invariant and simple valuation on Kn, which is also either monotone or continuous. Then it is a multiple of the volume.

Valuations on convex bodies The study of valuations, especially with continuity and invariance properties, is one of the most active and proliﬁc branches of convex geometry, after the seminal contributions by Hadwiger, McMullen and Alesker.

Thm. (Volume theorem). Let µ be a rigid motion invariant and simple valuation on Kn, which is also either monotone or continuous. Then it is a multiple of the volume.

Valuations on convex bodies

n n n Let K = {convex bodies in R }. A mapping µ : K → R is a valuation if

µ(K ∪ L) + µ(K ∩ L) = µ(K) + µ(L)

for every K and L s.t. K ∪ L ∈ Kn. Thm. (Volume theorem). Let µ be a rigid motion invariant and simple valuation on Kn, which is also either monotone or continuous. Then it is a multiple of the volume.

Valuations on convex bodies

n n n Let K = {convex bodies in R }. A mapping µ : K → R is a valuation if

µ(K ∪ L) + µ(K ∩ L) = µ(K) + µ(L)

n n n Let K = {convex bodies in R }. A mapping µ : K → R is a valuation if

µ(K ∪ L) + µ(K ∩ L) = µ(K) + µ(L)

Thm. (Volume theorem). Let µ be a rigid motion invariant and simple valuation on Kn, which is also either monotone or continuous. Then it is a multiple of the volume. Let X be a space of functions. A mapping σ : X → R is a valuation if

σ(f ∨ g) + σ(f ∧ g) = σ(f ) + σ(g)

for every f , g ∈ X s.t. f ∨ g and f ∧ g are in X .

I X = {deﬁnable functions}; Wright (’11), and Baryshnikov, Ghrist, Wright (’13). p n I X = L (R ) or X = Orlicz space; Tsang (’10, ’12) and Kone (’14). 1,p n n I X = W (R ) or X = BV (R ); Ludwig (’11, ’12, ’13), Ma (’15), Wang (’13).

Valuations on spaces of functions I X = {deﬁnable functions}; Wright (’11), and Baryshnikov, Ghrist, Wright (’13). p n I X = L (R ) or X = Orlicz space; Tsang (’10, ’12) and Kone (’14). 1,p n n I X = W (R ) or X = BV (R ); Ludwig (’11, ’12, ’13), Ma (’15), Wang (’13).

Valuations on spaces of functions

Let X be a space of functions. A mapping σ : X → R is a valuation if

σ(f ∨ g) + σ(f ∧ g) = σ(f ) + σ(g)

for every f , g ∈ X s.t. f ∨ g and f ∧ g are in X . p n I X = L (R ) or X = Orlicz space; Tsang (’10, ’12) and Kone (’14). 1,p n n I X = W (R ) or X = BV (R ); Ludwig (’11, ’12, ’13), Ma (’15), Wang (’13).

Valuations on spaces of functions

Let X be a space of functions. A mapping σ : X → R is a valuation if

σ(f ∨ g) + σ(f ∧ g) = σ(f ) + σ(g)

for every f , g ∈ X s.t. f ∨ g and f ∧ g are in X .

I X = {deﬁnable functions}; Wright (’11), and Baryshnikov, Ghrist, Wright (’13). 1,p n n I X = W (R ) or X = BV (R ); Ludwig (’11, ’12, ’13), Ma (’15), Wang (’13).

Valuations on spaces of functions

Let X be a space of functions. A mapping σ : X → R is a valuation if

σ(f ∨ g) + σ(f ∧ g) = σ(f ) + σ(g)

for every f , g ∈ X s.t. f ∨ g and f ∧ g are in X .

I X = {deﬁnable functions}; Wright (’11), and Baryshnikov, Ghrist, Wright (’13). p n I X = L (R ) or X = Orlicz space; Tsang (’10, ’12) and Kone (’14). Valuations on spaces of functions

Let X be a space of functions. A mapping σ : X → R is a valuation if

σ(f ∨ g) + σ(f ∧ g) = σ(f ) + σ(g)

for every f , g ∈ X s.t. f ∨ g and f ∧ g are in X .

I rigid motion invariant, i.e.

n µ(f ) = µ(f ◦ T ) ∀ T rigid motion of R ;

I simple (vanishes on functions which are a.e. zero);

I monotone decreasing;

I continuous (w.r.t. a suitable topology). Then µ can be written in the form Z µ(f ) = G(f )dx ∀ f Rn for a suitable function G (continuous and monotone).

A result for log-concave functions Then µ can be written in the form Z µ(f ) = G(f )dx ∀ f Rn for a suitable function G (continuous and monotone).

A result for log-concave functions

Thm. (Cavallina, C., ’15) Let σ be a valuation deﬁned on the space of log-concave functions, which is:

I rigid motion invariant, i.e.

n µ(f ) = µ(f ◦ T ) ∀ T rigid motion of R ;

I simple (vanishes on functions which are a.e. zero);

I monotone decreasing;

I continuous (w.r.t. a suitable topology). A result for log-concave functions

Thm. (Cavallina, C., ’15) Let σ be a valuation deﬁned on the space of log-concave functions, which is:

I rigid motion invariant, i.e.

n µ(f ) = µ(f ◦ T ) ∀ T rigid motion of R ;

I simple (vanishes on functions which are a.e. zero);

I monotone decreasing;

I continuous (w.r.t. a suitable topology). Then µ can be written in the form Z µ(f ) = G(f )dx ∀ f Rn for a suitable function G (continuous and monotone).