A sharp Santaló inequality in a slab with centrally symmetric sections
Abstract
We prove a sharp Santaló-type inequality for the product of V(φ) and V of the Legendre transform of φ, where φ is an even geometric convex function on ℝn and V(φ)=∫(1+φ)−(n+1). For this size functional, the Legendre transform and functional polarity give the same volume, so the same inequality holds for both dualities. Through a projective correspondence, the problem is equivalent to a new Santaló-type inequality with respect to standard volume and standard polarity, on a class of not necessarily centered convex bodies. This class consists of convex bodies K⊂ℝn×[−1,1] containing {0}×[−1,1], for which every horizontal section is centrally symmetric about the distinguished vertical axis. We determine the sharp upper bound for voln+1(K)voln+1(K∘) in this class and classify all equality cases. When n=1, the extremizers are the disk and its horizontal linear images. When n≥ 2, the ball is no longer a maximizer, and the extremizer is unique up to horizontal linear transformations and the reflection t↦ −t. The symmetric Santaló inequality applied to horizontal sections, together with monotone transport, reduces the problem to a maximization problem for increasing curves in the square [−1,1]2. We solve this problem using a global potential when n=1 and two Hamilton-Jacobi branches when n≥ 2. Finally, in dimension one we provide a sharp family of functional Santaló inequalities for the Legendre duality, interpolating between the volume V and the exponential volume ∫ exp(−φ), with centered quadratics as the only equality cases.
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BibTeX
@misc{artsteinavidan2026sharp,
title = {{A sharp Santaló inequality in a slab with centrally symmetric sections}},
author = {Shiri Artstein-Avidan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38041},
eprint = {2609.38041},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38041},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A sharp Santaló inequality in a slab with centrally symmetric sections
AU - Shiri Artstein-Avidan
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38041
UR - https://arxiv.org/abs/2609.38041
AB - We prove a sharp Santaló-type inequality for the product of V(φ) and V of the Legendre transform of φ, where φ is an even geometric convex function on ℝ^n and V(φ)=∫(1+φ)^−(n+1). For this size functional, the Legendre transform and functional polarity give the same volume, so the same inequality holds for both dualities. Through a projective correspondence, the problem is equivalent to a new Santaló-type inequality with respect to standard volume and standard polarity, on a class of not necessarily centered convex bodies. This class consists of convex bodies K⊂ℝ^n×[−1,1] containing {0}×[−1,1], for which every horizontal section is centrally symmetric about the distinguished vertical axis. We determine the sharp upper bound for vol_n+1(K)vol_n+1(K^∘) in this class and classify all equality cases. When n=1, the extremizers are the disk and its horizontal linear images. When n≥ 2, the ball is no longer a maximizer, and the extremizer is unique up to horizontal linear transformations and the reflection t↦ −t. The symmetric Santaló inequality applied to horizontal sections, together with monotone transport, reduces the problem to a maximization problem for increasing curves in the square [−1,1]^2. We solve this problem using a global potential when n=1 and two Hamilton-Jacobi branches when n≥ 2. Finally, in dimension one we provide a sharp family of functional Santaló inequalities for the Legendre duality, interpolating between the volume V and the exponential volume ∫ exp(−φ), with centered quadratics as the only equality cases.
ER -