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A Santaló inequality for the $L^p$-polar body

Vlassis Mastrantonis

Abstract

In recent work with Berndtsson and Rubinstein, a notion of $L^p$-polarity was introduced, with classical polarity recovered in the limit $p\to\infty$, and $L^1$-polarity closely related to Bergman kernels of tube domains. A Santaló inequality for the $L^p$-polar was proved for symmetric convex bodies. The aim of this article is to remove the symmetry assumption. Thus, an $L^p$-Santaló inequality holds for any convex body after translation by the $L^p$-Santaló point. As a corollary, this yields an optimal upper bound on Bergman kernels of tube domains. The proof is by Steiner symmetrization, but unlike the symmetric case, a careful translation of the body is required before each symmetrization.

A Santaló inequality for the $L^p$-polar body

Abstract

In recent work with Berndtsson and Rubinstein, a notion of -polarity was introduced, with classical polarity recovered in the limit , and -polarity closely related to Bergman kernels of tube domains. A Santaló inequality for the -polar was proved for symmetric convex bodies. The aim of this article is to remove the symmetry assumption. Thus, an -Santaló inequality holds for any convex body after translation by the -Santaló point. As a corollary, this yields an optimal upper bound on Bergman kernels of tube domains. The proof is by Steiner symmetrization, but unlike the symmetric case, a careful translation of the body is required before each symmetrization.
Paper Structure (10 sections, 22 theorems, 109 equations, 7 figures)

This paper contains 10 sections, 22 theorems, 109 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Inequality
  3. Outline of the proof of Theorem \ref{['LpSantaloThm']}
  4. Steiner symmetrization
  5. Symmetries of the Lᵖ-Santaló point
  6. Separating the Lᵖ-polar
  7. A generalization Ball's Brunn--Minkowski inequality for the harmonic mean
  8. Monotonicity of the volume of the slices of the Lᵖ-polar under Steiner symmetrization
  9. An inequality for the Lp-support function
  10. Finishing the proof

Key Result

Theorem 1.1

BMR Let $p\in (0,\infty]$. For a symmetric convex body $K\subset \mathbb{R}^n$, ${\mathcal{M}}_p(K)\leq {\mathcal{M}}_p(B_2^n)$.

Figures (7)

  • Figure 1: Steiner symmetrization could potentially decrease $L^p$-Mahler volume.
  • Figure 2: Translating $K$ in a direction normal to $u$ has the same effect on the Steiner symmetral.
  • Figure 3: Translating $K$ in a direction parallel to $u$ has no effect on the Steiner symmetral.
  • Figure 4: Translating $K$ in a direction parallel to $u$ so that $u^\perp$$1/2$-separates the $L^p$-polar.
  • Figure 5: If $K$ mostly lies in $(e_n^\perp)^+$, then $K^{\circ}$ mostly lies in $(e_n^\perp)^-$.
  • ...and 2 more figures

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Lemma 1.4
  • Corollary 1.5
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 38 more