TY - JOUR

T1 - Functional Brunn-Minkowski inequalities induced by polarity

AU - Artstein-Avidan, S.

AU - Florentin, D. I.

AU - Segal, A.

N1 - Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2020/4/15

Y1 - 2020/4/15

N2 - We prove a new family of inequalities, which compare the integral of a geometric convolution of non-negative functions with the integrals of the original functions. For classical inf-convolution, this type of inequality is called the Prékopa-Leindler inequality, which, restricted to indicators of convex bodies, gives the classical Brunn-Minkowski inequality. The convolution we consider is a different one, which arises from the study of the polarity transform for functions. While inf-convolution arises as the pull back of usual addition of convex functions under the Legendre transform, our geometric inf-convolution arises as the pull back of the second order reversing transform on geometric convex functions (called either polarity transform or A-transform). These are, up to linear terms, the only order reversing isomorphisms on the class of geometric convex functions. We prove that the integral of this new geometric convolution of two functions is bounded from below by the harmonic average of the individual integrals. Our inequality implies the Brunn-Minkowski inequality, as well as some other, new, inequalities for volumes of bodies. Our inequalities are intimately connected with Busemann's convexity theorem, a new variant of which we prove for 1-convex hulls and log-concave densities.

AB - We prove a new family of inequalities, which compare the integral of a geometric convolution of non-negative functions with the integrals of the original functions. For classical inf-convolution, this type of inequality is called the Prékopa-Leindler inequality, which, restricted to indicators of convex bodies, gives the classical Brunn-Minkowski inequality. The convolution we consider is a different one, which arises from the study of the polarity transform for functions. While inf-convolution arises as the pull back of usual addition of convex functions under the Legendre transform, our geometric inf-convolution arises as the pull back of the second order reversing transform on geometric convex functions (called either polarity transform or A-transform). These are, up to linear terms, the only order reversing isomorphisms on the class of geometric convex functions. We prove that the integral of this new geometric convolution of two functions is bounded from below by the harmonic average of the individual integrals. Our inequality implies the Brunn-Minkowski inequality, as well as some other, new, inequalities for volumes of bodies. Our inequalities are intimately connected with Busemann's convexity theorem, a new variant of which we prove for 1-convex hulls and log-concave densities.

KW - Concave functionals

KW - Convex functions

KW - Infimum convolution

KW - Integral inequalities

KW - Interpolation

UR - http://www.scopus.com/inward/record.url?scp=85078959519&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2020.107006

DO - 10.1016/j.aim.2020.107006

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AN - SCOPUS:85078959519

SN - 0001-8708

VL - 364

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107006

ER -