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On a Santaló point for Nakamura-Tsuji's Laplace transform inequality

Dario Cordero-Erausquin, Matthieu Fradelizi, Dylan Langharst

TL;DR

The paper develops a functional Blaschke–Santalo framework for the Laplace transform by introducing a $p$-Laplace transform $\mathcal{L}_p(f)$ and a centered functional volume product $M_p(f)$, proving that Gaussians maximize $M_p$ and that a unique Laplace–Santaló point $s_p(f)$ exists for general nonnegative $f$. A central technical achievement is showing that evolution under the Fokker–Planck/heat semigroups strictly improves the bound, with $M_p(f_t)$ increasing in time and bounded above by the Gaussian case, and the limit $p\to0^+$ recovers the Blaschke–Santalo inequality in its functional and essential-polar forms. The work unifies even and non-even cases, connects to polar transforms via double Laplace transforms, and yields a reversed hypercontractivity result in Gaussian settings, while also posing open questions about Santaló curves and centering dynamics. Overall, the paper advances a robust analytic route to functional geometric inequalities through centered Laplace transforms and semigroup flows, with potential implications for convex geometry and high-dimensional analysis.

Abstract

Nakamura and Tsuji recently obtained an integral inequality involving a Laplace transform of even functions that implies, at the limit, the Blaschke-Santaló inequality in its functional form. Inspired by their method, based on the Fokker-Planck semi-group, we extend the inequality to non-even functions. We consider a well-chosen centering procedure by studying the infimum over translations in a double Laplace transform. This requires a new look on the existing methods and leads to several observations of independent interest on the geometry of the Laplace transform. Application to reverse hypercontractivity is also given.

On a Santaló point for Nakamura-Tsuji's Laplace transform inequality

TL;DR

The paper develops a functional Blaschke–Santalo framework for the Laplace transform by introducing a -Laplace transform and a centered functional volume product , proving that Gaussians maximize and that a unique Laplace–Santaló point exists for general nonnegative . A central technical achievement is showing that evolution under the Fokker–Planck/heat semigroups strictly improves the bound, with increasing in time and bounded above by the Gaussian case, and the limit recovers the Blaschke–Santalo inequality in its functional and essential-polar forms. The work unifies even and non-even cases, connects to polar transforms via double Laplace transforms, and yields a reversed hypercontractivity result in Gaussian settings, while also posing open questions about Santaló curves and centering dynamics. Overall, the paper advances a robust analytic route to functional geometric inequalities through centered Laplace transforms and semigroup flows, with potential implications for convex geometry and high-dimensional analysis.

Abstract

Nakamura and Tsuji recently obtained an integral inequality involving a Laplace transform of even functions that implies, at the limit, the Blaschke-Santaló inequality in its functional form. Inspired by their method, based on the Fokker-Planck semi-group, we extend the inequality to non-even functions. We consider a well-chosen centering procedure by studying the infimum over translations in a double Laplace transform. This requires a new look on the existing methods and leads to several observations of independent interest on the geometry of the Laplace transform. Application to reverse hypercontractivity is also given.
Paper Structure (19 sections, 16 theorems, 190 equations)

This paper contains 19 sections, 16 theorems, 190 equations.

Table of Contents

  1. Introduction and main results
  2. Laplace transform, existence and continuity of the Laplace-Santaló point
  3. More on Laplace transform of log-concave functions
  4. Double Laplace transform and proof of Proposition \ref{['prop:sant']}
  5. The case $p=0$: the (essential) polar
  6. Continuity of the infimum and of the Laplace-Santaló point
  7. Semi-groups
  8. Fokker-Planck or heat semi-groups
  9. Preliminary Inequalities
  10. Proof of \ref{['eq:mainineq']} in Theorem \ref{['t:main']}
  11. Proof of \ref{['eq:monotone']} in Theorem \ref{['t:main']}
  12. Proof of \ref{['eq:bound_reg']} in Theorem \ref{['t:main']}
  13. Proof of Theorem \ref{['t:main_2']}, Theorem \ref{['t:main_sant']} and Theorem \ref{['t:main_sant_2']}
  14. Proof of Theorem \ref{['t:main_2']} and Theorem \ref{['t:main_sant']}
  15. Proof of Theorem \ref{['t:main_sant_2']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any $f\in L^p(\mathbb{R}^n)$ nonnegative, we have where $C_p=[p^\frac{1}{p}\left(-q\right)^{-\frac{1}{q}}]^{\frac{n}{2}} \, (2\pi)^{\frac{n}{q}}$ is so that there is equality when $f$ is a (centered) Gaussian function. If the supremum is finite, it is uniquely attained at some point $z$, and this point is zero if $\int_{\mathbb{R}^n} x\, L(f)^q(x) with equality when $f$ is a centered Gaussia

Theorems & Definitions (32)

  • Theorem 1.1: Reverse $L^p$ bound for the Laplace transform
  • Theorem 1.2
  • Proposition 1.3: $p$-Laplace-Santaló point
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • proof
  • proof
  • proof
  • ...and 22 more