Derivatives of integrals of log{concave functions Andrea Colesanti Convex and integral geometry Frankfurt, 26{30 September 2011 This is a report on an ongoing collaboration with Ilaria Fragal`a(Politecnico di Milano) n We consider functions f : R ! R+ of the form f = e−u ; where n u : R ! (−∞; 1] is convex. (with u 6≡ 1). We will always assume lim u(x) = 1 , lim f (x) = 0 : jxj!1 jxj!1 Examples 2 2 jxj f (x) = e−|xj =2 ; u(x) = : 2 1 if x 2 K, 0 if x 2 K, f (x) = χ (x) = u(x) = I (x) = K 0 if x 62 K, K 1 if x 62 K, n where K ⊂ R is a convex body. Log{concave functions We will always assume lim u(x) = 1 , lim f (x) = 0 : jxj!1 jxj!1 Examples 2 2 jxj f (x) = e−|xj =2 ; u(x) = : 2 1 if x 2 K, 0 if x 2 K, f (x) = χ (x) = u(x) = I (x) = K 0 if x 62 K, K 1 if x 62 K, n where K ⊂ R is a convex body. Log{concave functions n We consider functions f : R ! R+ of the form f = e−u ; where n u : R ! (−∞; 1] is convex. (with u 6≡ 1). , lim f (x) = 0 : jxj!1 Examples 2 2 jxj f (x) = e−|xj =2 ; u(x) = : 2 1 if x 2 K, 0 if x 2 K, f (x) = χ (x) = u(x) = I (x) = K 0 if x 62 K, K 1 if x 62 K, n where K ⊂ R is a convex body. Log{concave functions n We consider functions f : R ! R+ of the form f = e−u ; where n u : R ! (−∞; 1] is convex. (with u 6≡ 1). We will always assume lim u(x) = 1 jxj!1 Examples 2 2 jxj f (x) = e−|xj =2 ; u(x) = : 2 1 if x 2 K, 0 if x 2 K, f (x) = χ (x) = u(x) = I (x) = K 0 if x 62 K, K 1 if x 62 K, n where K ⊂ R is a convex body. Log{concave functions n We consider functions f : R ! R+ of the form f = e−u ; where n u : R ! (−∞; 1] is convex. (with u 6≡ 1). We will always assume lim u(x) = 1 , lim f (x) = 0 : jxj!1 jxj!1 2 2 jxj f (x) = e−|xj =2 ; u(x) = : 2 1 if x 2 K, 0 if x 2 K, f (x) = χ (x) = u(x) = I (x) = K 0 if x 62 K, K 1 if x 62 K, n where K ⊂ R is a convex body. Log{concave functions n We consider functions f : R ! R+ of the form f = e−u ; where n u : R ! (−∞; 1] is convex. (with u 6≡ 1). We will always assume lim u(x) = 1 , lim f (x) = 0 : jxj!1 jxj!1 Examples 1 if x 2 K, 0 if x 2 K, f (x) = χ (x) = u(x) = I (x) = K 0 if x 62 K, K 1 if x 62 K, n where K ⊂ R is a convex body. Log{concave functions n We consider functions f : R ! R+ of the form f = e−u ; where n u : R ! (−∞; 1] is convex. (with u 6≡ 1). We will always assume lim u(x) = 1 , lim f (x) = 0 : jxj!1 jxj!1 Examples 2 2 jxj f (x) = e−|xj =2 ; u(x) = : 2 0 if x 2 K, u(x) = I (x) = K 1 if x 62 K, n where K ⊂ R is a convex body. Log{concave functions n We consider functions f : R ! R+ of the form f = e−u ; where n u : R ! (−∞; 1] is convex. (with u 6≡ 1). We will always assume lim u(x) = 1 , lim f (x) = 0 : jxj!1 jxj!1 Examples 2 2 jxj f (x) = e−|xj =2 ; u(x) = : 2 1 if x 2 K, f (x) = χ (x) = K 0 if x 62 K, n where K ⊂ R is a convex body. Log{concave functions n We consider functions f : R ! R+ of the form f = e−u ; where n u : R ! (−∞; 1] is convex. (with u 6≡ 1). We will always assume lim u(x) = 1 , lim f (x) = 0 : jxj!1 jxj!1 Examples 2 2 jxj f (x) = e−|xj =2 ; u(x) = : 2 1 if x 2 K, 0 if x 2 K, f (x) = χ (x) = u(x) = I (x) = K 0 if x 62 K, K 1 if x 62 K, Log{concave functions n We consider functions f : R ! R+ of the form f = e−u ; where n u : R ! (−∞; 1] is convex. (with u 6≡ 1). We will always assume lim u(x) = 1 , lim f (x) = 0 : jxj!1 jxj!1 Examples 2 2 jxj f (x) = e−|xj =2 ; u(x) = : 2 1 if x 2 K, 0 if x 2 K, f (x) = χ (x) = u(x) = I (x) = K 0 if x 62 K, K 1 if x 62 K, n where K ⊂ R is a convex body. Let f and g be log{concave and s > 0. We set (f ⊕ g)(z) := sup f (x)g(y) ; x+y=z x (s · f )(x) := sf : s I These operations preserve log{concavity: f ⊕ g and s · f are log{concave. I If K and L are convex bodies and t; s > 0, then s · χK ⊕ t · χL = χsK+tL ; where sK + tL = fsx + ty : x 2 K ; y 2 Lg : Adding log{concave functions (Asplund sum) We set (f ⊕ g)(z) := sup f (x)g(y) ; x+y=z x (s · f )(x) := sf : s I These operations preserve log{concavity: f ⊕ g and s · f are log{concave. I If K and L are convex bodies and t; s > 0, then s · χK ⊕ t · χL = χsK+tL ; where sK + tL = fsx + ty : x 2 K ; y 2 Lg : Adding log{concave functions (Asplund sum) Let f and g be log{concave and s > 0. x (s · f )(x) := sf : s I These operations preserve log{concavity: f ⊕ g and s · f are log{concave. I If K and L are convex bodies and t; s > 0, then s · χK ⊕ t · χL = χsK+tL ; where sK + tL = fsx + ty : x 2 K ; y 2 Lg : Adding log{concave functions (Asplund sum) Let f and g be log{concave and s > 0. We set (f ⊕ g)(z) := sup f (x)g(y) ; x+y=z I These operations preserve log{concavity: f ⊕ g and s · f are log{concave. I If K and L are convex bodies and t; s > 0, then s · χK ⊕ t · χL = χsK+tL ; where sK + tL = fsx + ty : x 2 K ; y 2 Lg : Adding log{concave functions (Asplund sum) Let f and g be log{concave and s > 0. We set (f ⊕ g)(z) := sup f (x)g(y) ; x+y=z x (s · f )(x) := sf : s I If K and L are convex bodies and t; s > 0, then s · χK ⊕ t · χL = χsK+tL ; where sK + tL = fsx + ty : x 2 K ; y 2 Lg : Adding log{concave functions (Asplund sum) Let f and g be log{concave and s > 0. We set (f ⊕ g)(z) := sup f (x)g(y) ; x+y=z x (s · f )(x) := sf : s I These operations preserve log{concavity: f ⊕ g and s · f are log{concave. Adding log{concave functions (Asplund sum) Let f and g be log{concave and s > 0. We set (f ⊕ g)(z) := sup f (x)g(y) ; x+y=z x (s · f )(x) := sf : s I These operations preserve log{concavity: f ⊕ g and s · f are log{concave. I If K and L are convex bodies and t; s > 0, then s · χK ⊕ t · χL = χsK+tL ; where sK + tL = fsx + ty : x 2 K ; y 2 Lg : n ∗ Let u : R ! (−∞; 1] be convex. The conjugate function u of u is defined by u∗(y) = sup (x; y) − u(x) : n x2R ∗ n u : R ! (−∞; 1] is also a convex function. The ⊕ and · operations can be equivalently defined as follows. For f = e−u, g = e−v log{concave, and α; β > 0, α · f ⊕ β · g = α · e−u ⊕ β · e−v = e−w ; where w ∗ = αu∗ + βv ∗ : Hence ⊕ and · are linear, in the usual sense, with respect to the conjugates of the exponents (with reverse sign). Asplund sum and conjugate functions The conjugate function u∗ of u is defined by u∗(y) = sup (x; y) − u(x) : n x2R ∗ n u : R ! (−∞; 1] is also a convex function. The ⊕ and · operations can be equivalently defined as follows. For f = e−u, g = e−v log{concave, and α; β > 0, α · f ⊕ β · g = α · e−u ⊕ β · e−v = e−w ; where w ∗ = αu∗ + βv ∗ : Hence ⊕ and · are linear, in the usual sense, with respect to the conjugates of the exponents (with reverse sign). Asplund sum and conjugate functions n Let u : R ! (−∞; 1] be convex. ∗ n u : R ! (−∞; 1] is also a convex function. The ⊕ and · operations can be equivalently defined as follows. For f = e−u, g = e−v log{concave, and α; β > 0, α · f ⊕ β · g = α · e−u ⊕ β · e−v = e−w ; where w ∗ = αu∗ + βv ∗ : Hence ⊕ and · are linear, in the usual sense, with respect to the conjugates of the exponents (with reverse sign).