Abstract
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.
| Original language | English |
|---|---|
| Article number | 107006 |
| Journal | Advances in Mathematics |
| Volume | 364 |
| DOIs | |
| State | Published - 15 Apr 2020 |
| Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2020 Elsevier Inc.
Funding
Let p≥n+1 and let f,g,h:R>n→R+ be measurable functions such that ‖f‖p, ‖g‖p, and ‖h‖p are finite. Assume that [Formula presente] whenever t∈(0,1) and x,y∈Rn. Then ‖h‖p≥‖f‖p+‖g‖p. Acknowledgments: The authors would like to thank Bo'az Klartag for pointing out the relation between Busemann's theorem and the geometric λ-inf-convolution operation. We would also like to thank the anonymous referee for his helpful comments and for pointing out [16]. The first named author was supported in part by ISF grant No. 665/15 and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 770127). The second named author was supported in part by the U.S. National Science Foundation Grant DMS-1101636.
| Funders | Funder number |
|---|---|
| National Science Foundation | DMS-1101636 |
| Horizon 2020 Framework Programme | |
| Iowa Science Foundation | 665/15 |
| European Commission | |
| Horizon 2020 | 770127 |
Keywords
- Concave functionals
- Convex functions
- Infimum convolution
- Integral inequalities
- Interpolation