arXiv subjects

Hiroshi Tsuji

Publications and source records attributed to Hiroshi Tsuji.

13 recordsLinked to original sources

arXiv2026

On the monotonicity of magnitude functions of negative type finite semimetric spaces

Motivated by the conjectured monotonicity of the magnitude functions of metric spaces of negative type, we investigate the monotonicity problem for finite semimetric spaces, namely, ``metric spaces without the triangle inequality''. We completely resolve this problem by proving that the monotonicity holds for negative type semimetric spaces with at most three points, while constructing negative type semimetric spaces with at least four points whose magnitude functions are not monotonically increasing.

math.MG↗

arXiv2026

Operator capacity, the Brascamp--Lieb inequality and geometric programming

The capacity of completely positive operators and the Brascamp--Lieb constant can both be interpreted in terms of unconstrained geometric programming up to an additional minimisation over a compact group. We shine light on this perspective and make use of it to make novel contributions in both directions. For example, by making use of recent work of Bennett--Bez--Buschenhenke--Cowling--Flock, we prove new results regarding near-minimisers and local Hölder regularity of operator capacity. In addition, we observe that these results may be extended to the more general notion of capacity of quiver data. Furthermore, the geometric programming viewpoint allows us to give a new proof of the finiteness characterisation of the Brascamp--Lieb constant due to Bennett--Carbery--Christ--Tao (assuming Lieb's theorem on gaussian saturation).

math.FA↗

arXiv2026

The Gaussian Conjugate Rogers-Shephard Inequality

We fuse between the Rogers-Shephard inequality for the Lebesgue measure and Royen's Gaussian Correlation Inequality, simultaneously extending both into a single sharp inequality for the Gaussian measure $γ$ on $\mathbb{R}^n$, stating that \[ γ(K) γ(L) \leq γ(K\cap L) γ(K+L) \] whenever $K$ and $L$ are origin-symmetric convex sets in $\mathbb{R}^n$. This confirms a conjecture of M. Tehranchi [https://doi.org/10.1214/17-ECP89]. In fact, we show that the inequality remains valid whenever the Gaussian barycenters of $K$ and $L$ are at the origin, and characterize the equality cases. After rescaling, this also yields the following new inequality for convex sets with (Lebesgue) barycenters at the origin: \[ |K| |L| \leq |K \cap L| |K + L | ; \] this can be seen as a conjugate counterpart to Spingarn's extension of the Rogers-Shephard inequality (where $K+L$ is replaced by $K-L$ above). We also derive an additional conjugate version of a Gaussian inequality due to V. Milman and Pajor, as well as several extensions. Our main tool is a new Gaussian Forward-Reverse Brascamp-Lieb inequality for centered log-concave functions, of independent interest, which is crucially applicable to degenerate Gaussian covariances.

math.FA↗

arXiv2025

The Gaussian correlation inequality for centered convex sets and the case of equality

Inspired by Milman's recent observation, we prove that the Gaussian correlation inequality holds for convex sets having the same barycenter, and especially for centered ones. This gives an affirmative answer to the problem proposed by Szarek and Werner. We also characterize the equality case. The study of the equality case in the non-symmetric Gaussian correlation inequality relates to the following question: Let $X$ be a standard Gaussian random vector in $\mathbb{R}^n$. For which convex sets $K_1,K_2 \subset \mathbb{R}^n$, are the two events $\{X\in K_1\}$ and $\{X\in K_2\}$ independent? By imposing an additional normalization that $K_1$ and $K_2$ have the same barycenter, we give the necessary and sufficient conditions for this independence. The conditions also identify when $\|X\|_{K_1}$ and $\|X\|_{K_2}$ are independent as random variables.

math.FA↗

arXiv2025

Duality and Heat flow

We reveal the relation between the Legendre transform of convex functions and heat flow evolution, and how it applies to the functional Blaschke-Santalo inequality. We also describe local maximizers in this inequality.

math.FA↗

arXiv2024

A generalized Legendre duality relation and Gaussian saturation

Motivated by the barycenter problem in optimal transportation theory, Kolesnikov--Werner recently extended the notion of the Legendre duality relation for two functions to the case for multiple functions. We further generalize the duality relation and then establish the centered Gaussian saturation property for a Blaschke--Santaló type inequality associated with it. Our approach to the understanding such a generalized Legendre duality relation is based on our earlier observation that directly links Legendre duality with the inverse Brascamp--Lieb inequality. More precisely, for a large family of degenerate Brascamp--Lieb data, we prove that the centered Gaussian saturation property for the inverse Brascamp--Lieb inequality holds true when inputs are restricted to even and log-concave functions. As an application to convex geometry, we establish the most important case of a conjecture of Kolesnikov and Werner about the Blaschke--Santaló inequality for multiple even functions as well as multiple symmetric convex bodies. Furthermore, in the direction of information theory and optimal transportation theory, this provides an affirmative answer to another conjecture of Kolesnikov--Werner about a Talagrand type inequality for multiple even probability measures that involves the Wasserstein barycenter.

math.FA↗

arXiv2024

A note on ubiquity of geometric Brascamp-Lieb data

Relying substantially on work of Garg, Gurvits, Oliveira and Wigderson, it is shown that geometric Brascamp--Lieb data are, in a certain sense, ubiquitous. This addresses a question raised by Bennett and Tao in their recent work on the adjoint Brascamp--Lieb inequality.

math.CA↗

arXiv2024

The functional volume product under heat flow

We prove that the functional volume product for even functions is monotone increasing along the Fokker--Planck heat flow. This in particular yields a new proof of the functional Blaschke--Santaló inequality by K. Ball and also Artstein-Avidan--Klartag--Milman in the even case. This result is the consequence of a new understanding of the regularizing property of the Ornstein--Uhlenbeck semigroup. That is, we establish an improvement of Borell's reverse hypercontractivity inequality for even functions and identify the sharp range of the admissible exponents. As another consequence of successfully identifying the sharp range for the inequality, we derive the sharp $L^p$-$L^q$ inequality for the Laplace transform for even functions. The best constant of the inequality is attained by centered Gaussians, and thus this provides an analogous result to Beckner's sharp Hausdorff--Young inequality. Our technical novelty in the proof is the use of the Brascamp--Lieb inequality for log-concave measures and Cramér--Rao's inequality in this context.

math.FA↗

arXiv2023

Analytic aspects of the dilation inequality for symmetric convex sets in Euclidean spaces

We discuss an analytic form of the dilation inequality for symmetric convex sets in Euclidean spaces, which is a counterpart of analytic aspects of Cheeger's isoperimetric inequality. We show that the dilation inequality for symmetric convex sets is equivalent to a certain bound of the relative entropy for symmetric quasi-convex functions, which is close to the logarithmic Sobolev inequality or Cramér--Rao inequality. As corollaries, we investigate the reverse Shannon inequality, logarithmic Sobolev inequality, Kahane--Khintchine inequality, deviation inequality and isoperimetry. We also give new probability measures satisfying the dilation inequality for symmetric convex sets via bounded perturbations and tensorization.

math.MG↗

arXiv2022

Hypercontractivity beyond Nelson's time and its applications to Blaschke--Santaló inequality and inverse Santaló inequality

We explore an interplay between an analysis of diffusion flows such as Ornstein--Uhlenbeck flow and Fokker--Planck flow and inequalities from convex geometry regarding the volume product. More precisely, we introduce new types of hypercontractivity for the Ornstein--Uhlenbeck flow and clarify how these imply the Blaschke--Santaló inequality and the inverse Santaló inequality, also known as Mahler's conjecture. Motivated the link, we establish two types of new hypercontractivity in this paper. The first one is an improvement of Borell's reverse hypercontractivity inequality in terms of Nelson's time relation under the restriction that the inputs have an appropriate symmetry. We then prove that it implies the Blaschke--Santaló inequality. At the same time, it also provides an example of the inverse Brascamp--Lieb inequality due to Barthe--Wolff beyond their non-degenerate condition. The second one is Nelson's forward hypercontractivity inequality with exponents below 1 for the inputs which are log-convex and semi-log-concave. This yields new lower bounds of the volume product for convex bodies whose boundaries are well curved. This consequence provides a quantitative result of works by Stancu and Reisner--Schütt--Werner where they observed that a convex body with well curved boundary is not a local minimum of the volume product.

math.MG↗

arXiv2022

Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality

We provide deficit estimates for Nelson's hypercontractivity inequality, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality under the restriction that the inputs are semi-log-subharmonic, semi-log-convex, or semi-log-concave. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan, Lehec and Shenfeld concerning a deficit estimate for inputs with small covariance. Similarly, our result on Talagrand's transportation cost inequality complements and, for a large class of semi-log-concave inputs, improves a deficit estimate recently proved by Mikulincer. Our deficit estimates for hypercontractivity will be obtained by using a flow monotonicity scheme built on the Fokker--Planck equation, and our deficit estimates for the logarithmic Sobolev inequality will be derived as a corollary. For Talagrand's inequality, we use an optimal transportation argument. An appealing feature of our framework is robustness and this allows us to derive deficit estimates for the hypercontracivity inequality associated with the Hamilton--Jacobi equation, the Poincaré inequality, and for Beckner's inequality.

math.AP↗

arXiv2021

Dilation type inequalities for strongly-convex sets in weighted Riemannian manifolds

In this paper, we consider a dilation type inequality on a weighted Riemannian manifold, which is classically known as Borell's lemma in high-dimensional convex geometry. We investigate the dilation type inequality as an isoperimetric type inequality by introducing the dilation profile and estimate it by the one for the corresponding model space under lower weighted Ricci curvature bounds. We also explore functional inequalities derived from the comparison of the dilation profiles under the nonnegative weighted Ricci curvature. In particular, we show several functional inequalities related to various entropies.

math.DG↗

arXiv2020

Symmetrized Talagrand Inequalities on Euclidean Spaces

In this paper, we study the symmetrized Talagrand inequality that was proved by Fathi and has a connection with the Blaschke-Santaló inequality in convex geometry. As corollaries of our results, we have several refined functional inequalities under some conditions. We also give an alternative proof of Fathi's symmetrized Talagrand inequality on the real line and some applications.

math.DG↗