arXiv subjects

Matthieu Fradelizi

Publications and source records attributed to Matthieu Fradelizi.

At least 19 recordsLinked to original sources

arXiv2026

Volume and Projection Inequalities II: Determinants and $L_p$-Sums

We study inequalities for the volume of orthogonal projections and their relation to Firey $L_p$-sum, together with their determinant-power analogues, motivated by the Dembo--Cover--Thomas conjecture. For $L_p$-zonoids $K,L\subset\mathbb{R}^n$ and $u\in S^{n-1}$, we consider the inequality \[ \left( \frac{|K\oplus_p L|} {|P_{u^\perp}(K\oplus_p L)|} \right)^p \geq \left( \frac{|K|}{|P_{u^\perp}K|} \right)^p + \left( \frac{|L|}{|P_{u^\perp}L|} \right)^p . \] For every $1<p<2$, we prove that this inequality fails in every dimension $n\geq2$. In contrast, the weak one-term inequality, obtained by omitting the second term on the right-hand side, holds in dimension two throughout the full range $1\leq p\leq2$. The proof of this planar result uses a sharp estimate for the normalized duality map. We also classify the corresponding determinant-power inequalities in the range $0<p<2$. The strong two-term inequality holds in dimension two and fails in every dimension $n\geq3$. The weak one-term inequality holds for $0<p\leq1$ in dimensions $n\leq3$ and fails for $n\geq4$; for $1<p<2$, it holds only in dimension two.

math.MG↗

arXiv2026

Volume and Projection Inequalities I: Zonoids and Courtade's Conjecture

We study volume and projection inequalities for zonoids through the multiaffine determinant polynomials that encode their volumes. We show that a log-submodularity conjecture for the volume of zonoids is equivalent to the Rayleigh property of zonotope volume polynomials, which we prove for the case when the degree or codegree is at most 3. This result is sharp in that when the degree and codegree are at least 4, we construct counterexamples using the existence of non-Rayleigh matroids in ranks at least four. Additionally, we provide unimodular or graphical counterexamples in dimensions four and higher to an equivalent projection inequality formulation of the conjecture. We also show that a stronger projection inequality fails already in dimension three. We next disprove Courtade's conjecture using a pair of orthogonal double bodies of revolution. Although Courtade's conjecture was originally formulated for general convex bodies, we show that it fails even for zonoids in every dimension at least three.

math.MG↗

arXiv2026

Entropic analogues of Grünbaum's inequality

The classical Grünbaum inequality asserts that the proportion of the volume of a convex body cut off by a halfspace containing its barycenter is at least $1/e$. From its functional counterpart, for any log-concave random variable $X$, one has $\mathbb{P}(X\ge \mathbb{E}X)\ge 1/e$, with equality if and only if $X$ is exponential. Motivated by Grünbaum's inequality for convex bodies and its functional generalizations, we prove analogous inequalities for entropy, with characterizations of the equality cases. We show that if $X$ is a log-concave random variable on $\mathbb{R}$, then $$ h(X)-\frac{e}{e-1}H_2(1/e) \leq h(X|X \leq \mathbb{E}X) \leq h(X), $$ where $h$ is the differential entropy, $H_2(\cdot)$ is the binary entropy function and $X|X\leq \mathbb{E}X$ stands for the distribution of $X$ conditional on $X\leq \mathbb{E}X$. We generalize the upper bound for all Rényi entropies and the lower bound for min-entropy. Our inequalities are sharp and we characterize all equality cases. We discuss potential generalizations in high dimensions and give counterexamples in some directions. As an intermediate step for the proof of the lower bound, we establish a new inequality that we prove using a technique known as degrees of freedom, combined with a standard KKT-type optimization lemma. Along the way, we characterize the equality case in a known comparison inequality between differential and min-entropy, which may be of independent interest.

math.PR↗

arXiv2026

L_p-Rogers--Shephard type inequalities for L_p-zonoids and symmetric bodies

We study generalizations of the classical Rogers--Shephard inequalities in the framework of Firey $L_p$-summation. We first consider the class of asymmetric $L_p$-zonoids. In this setting, we show that proving a sharp $L_p$-Rogers--Shephard inequality for asymmetric $L_p$-zonoids in $\mathbb{R}^n$ is equivalent to proving a sharp inequality between the volumes of projections of $B_q^m\cap \mathbb{R}^m_+$ and $B_q^m$ onto an $n$-dimensional subspace $E$, where $q$ is the Hölder conjugate of $p$. We conjecture that the inequality is sharp when the subspace $E$ is a coordinate subspace. We fully establish this inequality along with equality conditions in the case $p =2$. For general $p$, we prove it in the case $n=m-1$, $n=1$, and discuss several particular cases, including an averaged version and a local version of the inequality. We then turn to the setting of convex bodies having a center of symmetry. Rogers and Shephard also proved a sharp version of their inequality for bodies in this class. We conjecture a similar bound for the $L_p$-summation, and we establish our conjecture for the particular case of asymmetric $L_1$-zonoids, which, in particular, proves our conjecture in the planar case.

math.MG↗

arXiv2026

Equality cases for the $L_p$-Rogers--Shephard inequality in the plane and for locally anti-blocking bodies in $\mathbb{R}^n$

The classical Rogers--Shephard inequalities were extended to the Firey $L_p$-summation by Bianchini and Colesanti in the plane and by Zvavitch and the second and fourth authors for locally anti-blocking convex bodies in $\mathbb{R}^n$, leaving open the equality cases. We characterize the equality cases of these inequalities: in both cases, for $p>1$, equality holds if and only if the convex body is a simplex with one vertex at the origin.

math.MG↗

arXiv2026

Weighted Brunn-Minkowski Theory II: Inequalities for Mixed Measures and Applications

In "Weighted Brunn-Minkowski Theory I", the prequel to this work, we discussed how recent developments on concavity of measures have laid the foundations of a nascent weighted Brunn-Minkowski theory. In particular, we defined the mixed measures of three convex bodies and obtained its integral representation. In this work, we obtain inequalities for mixed measures, such as a generalization of Fenchel's inequality; this provides a new, simpler proof of the classical volume case. Moreover, we show that mixed measures are connected to the study of log-submodularity and supermodularity of the measure of Minkowski sums of convex bodies. This elaborates on the recent investigations of these properties for the Lebesgue measure. We conclude by establishing that the only Radon measures that are supermodular over the class of compact, convex sets are multiples of the Lebesgue measure. Motivated by this result, we then discuss weaker forms of supermodularity by restricting the class of convex sets.

math.FA↗

arXiv2025

On the monotonicity of discrete entropy for log-concave random vectors on $\mathbb{Z}^d$

We prove the following type of discrete entropy monotonicity for sums of isotropic, log-concave, independent and identically distributed random vectors $X_1,\dots,X_{n+1}$ on $\mathbb{Z}^d$: $$ H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n}) + \frac{d}{2}\log{\Bigl(\frac{n+1}{n}\Bigr)} +o(1), $$ where $o(1)$ vanishes as $H(X_1) \to \infty$. Moreover, for the $o(1)$-term, we obtain a rate of convergence $ O\Bigl({H(X_1)}{e^{-\frac{1}{d}H(X_1)}}\Bigr)$, where the implied constants depend on $d$ and $n$. This generalizes to $\mathbb{Z}^d$ the one-dimensional result of the second named author (2023). As in dimension one, our strategy is to establish that the discrete entropy $H(X_1+\cdots+X_{n})$ is close to the differential (continuous) entropy $h(X_1+U_1+\cdots+X_{n}+U_{n})$, where $U_1,\dots, U_n$ are independent and identically distributed uniform random vectors on $[0,1]^d$ and to apply the theorem of Artstein, Ball, Barthe and Naor (2004) on the monotonicity of differential entropy. In fact, we show this result under more general assumptions than log-concavity, which are preserved up to constants under convolution. In order to show that log-concave distributions satisfy our assumptions in dimension $d\ge2$, more involved tools from convex geometry are needed because a suitable position is required. We show that, for a log-concave function on $\mathbb{R}^d$ in isotropic position, its integral, barycenter and covariance matrix are close to their discrete counterparts. Moreover, in the log-concave case, we weaken the isotropicity assumption to what we call almost isotropicity. One of our technical tools is a discrete analogue to the upper bound on the isotropic constant of a log-concave function, which extends to dimensions $d\ge1$ a result of Bobkov, Marsiglietti and Melbourne (2022).

math.PR↗

arXiv2025

Grünbaum's inequality for Gaussian and convex probability measures

A celebrated result in convex geometry is Grünbaum's inequality, which quantifies how much volume of a convex body can be cut off by a hyperplane passing through its barycenter. In this work, we establish a series of sharp Grünbaum-type inequalities - with equality characterizations - for probability measures under certain concavity assumptions. As an application, we apply the renowned Ehrhard inequality and deduce an ``Ehrhard-Grünbaum'' inequality for the Gaussian measure on $\mathbb{R}^n$, which improves upon the bound derived from its log-concavity. For $s$-concave Radon measures, our framework provides a simpler proof of known results and, more importantly, yields the previously missing equality characterization. This is achieved by gaining new insight into the equality case of their Brunn-Minkowski-type inequality. Moreover, we show that these ``$s$-Grünbaum'' inequalities can hold only when $s > -1$. However, for convex measures on the real line, we prove Grünbaum-type inequalities involving their cumulative distribution function.

math.FA↗

arXiv2025

On Mahler's conjecture for even s-concave functions in dimensions 1 and 2

In this paper, we establish different sharp forms of Mahler's conjecture for $s$-concave even functions in dimensions $n$, for $n=1$ and $2$, for $s>-1/n$, thus generalizing our previous results in \cite{FN} on log-concave even functions in dimension 2, which corresponds to the case $s=0$. The functional volume product of an even $s$-concave function $g$ is \[ \int_{\mathbb{R}^{n}}g(x)dx\int_{\mathbb{R}^{n}}\mathcal{L}_{s}g(y)dy, \] where $\mathcal{L}_{s}g$ is the $s$-polar function associated to $g$. The analogue of Mahler's conjecture for even $s$-concave functions postulates that this quantity is minimized for the indicatrix of a cube for any $s>-1/n$. In dimension $n=1$, we prove this conjecture for all $s\in(-1,0)$ (the case $s\ge0$ was established by the first author and Mathieu Meyer in \cite[page 17]{FM10}). In dimension $n=2$, we only consider the case $1/s\in\mathbb{Z}$: for $s>0$, we establish Mahler's conjecture for general $s$-concave even functions; for $s<0$, the situation is more involved, we only prove a sharp inequality for $s$-concave functions $g$ such that $g^s$ admits an asymptote in every direction. Notice that this set of functions is quite natural to consider, when $s<0$, since it is the largest subset of $s$-concave functions stable by $s$-duality.

math.FA↗

arXiv2025

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

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.

math.FA↗

arXiv2025

Entropic versions of Bergström's and Bonnesen's inequalities

We establish analogues of the Bergström and Bonnesen inequalities, related to determinants and volumes respectively, for the entropy power and for the Fisher information. The obtained inequalities strengthen the well-known convolution inequality for the Fisher information as well as the entropy power inequality in dimensions $d>1$, while they reduce to the former in $d=1$. Our results recover the original Bergström inequality and generalize a proof of Bergström's inequality given by Dembo, Cover and Thomas. We characterize the equality case in our entropic Bonnesen inequality.

cs.IT↗

arXiv2024

Entropy, slicing problem and functional Mahler's conjecture

In a recent work, Bo'az Klartag showed that, given a convex body with minimal volume product, its isotropic constant is related to its volume product. As a consequence, he obtained that a strong version of the slicing conjecture implies Mahler's conjecture. In this work, we extend these geometrical results to the realm of log-concave functions. In this regard, the functional analogues of the projective perturbations of the body are the log-Laplace perturbations of the function. The differentiation along these transformations is simplified thanks to the known properties of the log-Laplace transform. Moreover, we show that achieving such an analogous result requires the consideration of the suitable version of the isotropic constant, notably the one incorporating the entropy. Finally, an investigation into the equivalences between the functional and geometrical strong forms of the slicing conjecture is provided.

math.MG↗

arXiv2024

Some obstructions to contraction theorems on the half-sphere

Caffarelli's contraction theorem states that probability measures with uniformly logconcave densities on R d can be realized as the image of a standard Gaussian measure by a globally Lipschitz transport map. We discuss some counterexamples and obstructions that prevent a similar result from holding on the half-sphere endowed with a uniform measure, answering a question of Beck and Jerison.

math.FA↗

arXiv2023

Transport-entropy forms of direct and Converseblaschke-Santal{ó} inequalities

We explore alternative functional or transport-entropy formulations of the Blaschke-Santal{ó} inequality and of its conjectured counterpart due to Mahler. In particular, we obtain new direct and reverse Blaschke-Santal{ó} inequalities for s-concave functions. We also obtain new sharp symmetrized transport-entropy inequalities for a large class of spherically invariant probability measures, including the uniform measure on the unit Euclidean sphere and generalized Cauchy and Barenblatt distributions. Finally, we show that the Mahler's conjecture is equivalent to some reinforced log-Sobolev type inequality on the sphere.

math.FA↗

arXiv2023

Weighted Brunn-Minkowski Theory I: On Weighted Surface Area Measures

The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of these concepts to Borel measures with density in $\mathbb{R}^n$-- in particular, the weighted versions of mixed volumes (the so-called mixed measures) when dealing with up to three distinct convex bodies. We then formulate and analyze weighted versions of classical surface area measures, and obtain a new integral formula for the mixed measure of three bodies. As an application, we prove a Bézout-type inequality for rotational invariant log-concave measures, generalizing a result by Artstein-Avidan, Florentin and Ostrover. The results are new and interesting even for the special case of the standard Gaussian measure.

math.MG↗

arXiv2023

The functional form of Mahler conjecture for even log-concave functions in dimension $2$

Let $Φ$ : R n $\rightarrow$ R $\cup$ {+$\infty$} be an even convex function and L$Φ$ be its Legendre transform. We prove the functional form of Mahler conjecture concerning the functional volume product P ($Φ$) = e --$Φ$ e --L$Φ$ in dimension 2: we give the sharp lower bound of this quantity and characterize the equality case. The proof uses the computation of the derivative in t of P (t$Φ$) and ideas due to Meyer [M] for unconditional convex bodies, adapted to the functional case by Fradelizi-Meyer [FM2] and extended for symmetric convex bodies in dimension 3 by Iriyeh-Shibata [IS] (see also [FHMRZ]).

math.FA↗

arXiv2023

Volume Product

Our purpose here is to give an overview of known results and open questions concerning the volume product ${\mathcal P}(K)=\min_{z\in K}{\rm vol}(K){\rm vol}((K-z)^*)$ of a convex body $K$ in ${\mathbb R}^n$. We present a number of upper and lower bounds for ${\mathcal P}(K)$, in particular, we discuss the Mahler's conjecture on the lower bound of ${\mathcal P}(K)$, which is still open. We also show connections of ${\mathcal P}(K)$ with different parts of modern mathematics, including Geometric Number Theory, Convex Geometry, Analysis, Harmonic Analysis as well as Systolic and Symplectic Geometries and Probability.

math.MG↗

arXiv2022

On the volume of the Minkowski sum of zonoids

We explore some inequalities in convex geometry restricted to the class of zonoids. We show the equivalence, in the class of zonoids, between a local Alexandrov-Fenchel inequality, a local Loomis-Whitney inequality, the log-submodularity of volume, and the Dembo-Cover-Thomas conjecture on the monotonicity of the ratio of volume to the surface area. In addition to these equivalences, we confirm these conjectures in ${\mathbb R}^3$ and we establish an improved inequality in ${\mathbb R^2}$. Along the way, we give a negative answer to a question of Adam Marcus regarding the roots of the Steiner polynomial of zonoids. We also investigate analogous questions in the $L_p$-Brunn-Minkowski theory, and in particular, we confirm all of the above conjectures in the case $p=2$, in any dimension.

math.MG↗