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 [ f+1g is convex (e = 0). We will always assume lim u(x) = +1 ) lim f (x) = 0; jxj!+1 jxj!+1 and set n n C = ff log-concave on R : limjxj!+1 f (x) = 0g: Log-concave functions (e−∞ = 0). We will always assume lim u(x) = +1 ) lim f (x) = 0; jxj!+1 jxj!+1 and set n n C = ff log-concave on R : limjxj!+1 f (x) = 0g: Log-concave functions A function n f : R ! R+ is said to be log-concave if f = e−u where n u : R ! R [ f+1g is convex We will always assume lim u(x) = +1 ) lim f (x) = 0; jxj!+1 jxj!+1 and set n n C = ff log-concave on R : limjxj!+1 f (x) = 0g: Log-concave functions A function n f : R ! R+ is said to be log-concave if f = e−u where n −∞ u : R ! R [ f+1g is convex (e = 0). and set n n C = ff log-concave on R : limjxj!+1 f (x) = 0g: Log-concave functions A function n f : R ! R+ is said to be log-concave if f = e−u where n −∞ u : R ! R [ f+1g is convex (e = 0). We will always assume lim u(x) = +1 ) lim f (x) = 0; jxj!+1 jxj!+1 Log-concave functions A function n f : R ! R+ is said to be log-concave if f = e−u where n −∞ u : R ! R [ f+1g is convex (e = 0). We will always assume lim u(x) = +1 ) lim f (x) = 0; jxj!+1 jxj!+1 and set n n C = ff log-concave on R : limjxj!+1 f (x) = 0g: 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 2 K, f (x) = χ (x) = K 0 if x 2= K. Note that f = e−IK where 0 if x 2 K I (x) = = indicatrix function of K. K +1 if x 2= K Examples I The characteristic function of a convex body K (compact n convex subset of R ): 1 if x 2 K, f (x) = χ (x) = K 0 if x 2= K. Note that f = e−IK where 0 if x 2 K I (x) = = indicatrix function of K. K +1 if x 2= K Examples I The Gaussian function: 2 f (x) = e−kxk =2: Note that f = e−IK where 0 if x 2 K I (x) = = indicatrix function of K. K +1 if x 2= 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 2 K, f (x) = χ (x) = K 0 if x 2= 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 2 K, f (x) = χ (x) = K 0 if x 2= K. Note that f = e−IK where 0 if x 2 K I (x) = K +1 if x 2= 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 2 K, f (x) = χ (x) = K 0 if x 2= K. Note that f = e−IK where 0 if x 2 K I (x) = = indicatrix function of K. K +1 if x 2= 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... 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 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... 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... 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... 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. For f ; g 2 Cn we set (f ⊕ g)(z) = supff (x)g(y) j x + y = zg : 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 2 Cn we set (f ⊕ g)(z) = supff (x)g(y) j x + y = zg : (so that, e.g., f ⊕ f = 2 · f ). Cn is closed with respect to these operations. Addition of log-concave functions For f ; g 2 Cn we set (f ⊕ g)(z) = supff (x)g(y) j x + y = zg : 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 2 Cn we set (f ⊕ g)(z) = supff (x)g(y) j x + y = zg : 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 2 Cn we set (f ⊕ g)(z) = supff (x)g(y) j x + y = zg : 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 = fαx + βy : x 2 K; y 2 Lg: 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.