arXiv subjects

Francisco Marín Sola

Publications and source records attributed to Francisco Marín Sola.

9 recordsLinked to original sources

arXiv2026

Affine variational functionals, isoperimetric problems and stability

We study three affine variational quantities built on the affine $L^p$ energy of Lutwak, Yang and Zhang: the affine Poincaré--Sobolev constants $Λ^{\mathcal{A}}_{p,q}(Ω)$ of a bounded open set, the associated affine torsional rigidity, and the affine $p$-capacity of a compact set. We show that ellipsoids are the only extremal sets in their corresponding isoperimetric inequalities, and that the affine capacity of a compact set of positive volume is attained by a potential solving the affine $p$-Laplace equation. We also prove an affine Kohler-Jobin inequality: among sets with prescribed affine torsional rigidity, only ellipsoids minimize $Λ^{\mathcal{A}}_{p,q}$. Following the approach of Haddad, Jiménez and Montenegro to affine Sobolev inequalities, our methods rest on the $L_p$ Busemann--Petty centroid inequality. Proving a stability version of it, we obtain quantitative forms of the isoperimetric inequalities for $Λ^{\mathcal{A}}_{p,q}$, the affine torsional rigidity and the affine $p$-capacity, together with a stability result for the affine Kohler-Jobin inequality on convex sets.

math.AP↗

arXiv2026

On stochastic forms of functional isoperimetric inequalities

We present a probabilistic interpretation of several functional isoperimetric inequalities within the class of $p$-concave functions, building on random models for such functions introduced by P. Pivovarov and J. Rebollo-Bueno. First, we establish a stochastic isoperimetric inequality for a functional extension of the classical quermassintegrals, which yields a Sobolev-type inequality in this random setting as a particular case. Motivated by the latter, we further show that Zhang's affine Sobolev inequality holds in expectation when dealing with these random models of $p$-concave functions. Finally, we confirm that our results recover both their geometric analogues and deterministic counterparts. As a consequence of the latter, we establish a generalization of Zhang's affine Sobolev inequality restricted to $p$-concave functions in the context of convex measures.

math.FA↗

arXiv2026

Remarks on Brunn-Minkowski-type inequalities related to the Ornstein-Uhlenbeck operator

We investigate Brunn-Minkowski-type inequalities for the torsional rigidity $T_γ$ and the first eigenvalue $λ_γ$ associated with the Ornstein-Uhlenbeck operator. Counterexamples are provided showing that neither concavity nor convexity properties hold for $T_γ$ on general bounded convex sets. We also demonstrate that log-concavity and log-convexity properties fail in this setting. In the case of centrally symmetric sets, we answer a question raised by Cordero-Erausquin and Eskenazis by showing that $T_γ^{1/(n+2)}$ is neither convex nor concave. On the positive side, we prove that $T_γ^{1/3}$ is convex with respect to Minkowski addition when restricted to Euclidean balls centered at the origin. For $λ_γ$, we answer negatively a question posed by Colesanti, Francini, Livshyts, and Salani by showing that the inequality $λ_γ(Ω_t)^{-1/2} \geq (1-t)λ_γ(Ω_0)^{-1/2} + tλ_γ(Ω_1)^{-1/2}$ does not hold, even for centrally symmetric sets.

math.AP↗

arXiv2025

Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions

The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric-type inequalities for convex bodies. Following a classical work by Schneider, both inequalities have been extended to the so-called $m$th-order setting. In this work, we establish the $m$th-order analogues for these inequalities in the setting of log-concave functions. Our proof of the functional Zhang's projection inequality employs properties of the asymmetric LYZ body, significantly streamlining the argument and producing a novel approach for the case $m=1$. Furthermore, we introduce and analyze the radial mean bodies of a log-concave function, thereby providing a functional generalization of Gardner and Zhang's radial mean bodies. These are new even in the case $m=1$. Our development leverages an extension of Ball bodies, which may be of independent interest.

math.MG↗

arXiv2025

Quantitative improvements of functional inequalities under concavity properties

A classical result of Hensley provides a sharp lower bound for the functional $\int_\mathbb{R} t^2f$, where $f$ is a non-negative, even log-concave function. In the context of studying the minimal slabs of the unit cube, Barthe and Koldobsky established a quantitative improvement of Hensley's bound. In this work, we complement their result in several directions. First, we prove the corresponding upper bound inequality for $s$-concave functions with $s\geq 0$. Second, we present a generalization of Barthe and Koldobsky's result for functionals of the form $\int_\mathbb{R} Nf\,\mathrm{d}μ$, where $N$ is a convex, even function and $μ$ belongs to a suitable class of positive Borel measures. As a consequence of the employed methods, we obtain quantitative refinements of classical inequalities for $p$-norms and for the entropy of log-concave functions. Finally, we discuss both geometric consequences and probabilistic interpretations of our results.

math.FA↗

arXiv2025

On general versions of the Petty projection inequality

The classical Petty projection inequality is an affine isoperimetric inequality which constitutes a cornerstone in the affine geometry of convex bodies. By extending the polar projection body to an inter-dimensional operator, Petty's inequality was generalized to the so-called $(L_p,Q)$ setting, where $Q$ is an $m$-dimensional compact convex set. In this work, we further extend the $(L_p,Q)$ Petty projection inequality to the broader realm of rotationally invariant measures with concavity properties, namely, those with $γ$-concave density (for $γ\geq-1/nm$). Moreover, when $p=1$, and motivated by a contemporary empirical reinterpretation of Petty's result, we explore empirical analogues of this inequality.

math.MG↗

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↗

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

A general functional version of Grünbaum's inequality

A classical inequality by Grünbaum provides a sharp lower bound for the ratio $\mathrm{vol}(K^{-})/\mathrm{vol}(K)$, where $K^{-}$ denotes the intersection of a convex body with non-empty interior $K\subset\mathbb{R}^n$ with a halfspace bounded by a hyperplane $H$ passing through the centroid $\mathrm{g}(K)$ of $K$. In this paper we extend this result to the case in which the hyperplane $H$ passes by any of the points lying in a whole uniparametric family of $r$-powered centroids associated to $K$ (depending on a real parameter $r\geq0$), by proving a more general functional result on concave functions. The latter result further connects (and allows one to recover) various inequalities involving the centroid, such as a classical inequality (due to Minkowski and Radon) that relates the distance of $\mathrm{g}(K)$ to a supporting hyperplane of $K$, or a result for volume sections of convex bodies proven independently by Makai Jr. & Martini and Fradelizi.

math.MG↗