arXiv subjects

Dylan Langharst

Publications and source records attributed to Dylan Langharst.

At least 19 recordsLinked to original sources

arXiv2026

On cost-induced Santal\'o-type inequalities in Polish measure spaces

We introduce a framework for establishing Blaschke-Santal\'o-type inequalities on $m$-tuples of Polish measure spaces coupled together by a continuous cost function. Central to our approach is a transference principle, which provides a mechanism to lift geometric weighted inequalities involving cost-polar sets into functional integral inequalities of Santal\'o-type. We call these equivalent inequalities cost-Santal\'o inequalities. This definition expands and includes previous notions in the literature. We apply this principle to deduce several new versions of functional Santal\'o inequalities, including on the space of rectangular matrices and a functional sine Santal\'o inequality. A surprising development is that probability spaces with log-concave isoperimetric functions fit into our framework, for example, Gauss space and spherical space, leading to new functional Santal\'o inequalities in these settings. In particular, we obtain results for $\operatorname{RCD}(K,\infty)$ spaces. As a discrete application, we obtain an inequality for the Hamming cube. Finally, we explore applications to optimal transport, utilizing our functional framework to establish generalized transport-entropy inequalities on arbitrary Polish spaces satisfying a cost-Santal\'o inequality, which we explicitly instantiate for matrix spaces.

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arXiv2026

Convexity of Radial Mean Bodies via an Extension of Ball's Bodies

In this work, we extend a classical theorem of Keith Ball on integrals of log-concave functions along rays against the weight $r^{p-1}$ to the previously inaccessible regime $p\in (-1,0)$: if $g:\mathbb R^n\to\mathbb R_+$ is an integrable, upper semi-continuous, log-concave function which attains its maximum at the origin, then \[ x\mapsto \left(\frac{p}{g(o)}\int_{0}^{\infty}r^{p-1}(g(rx)-g(o))\mathrm{d}\,r\right)^{-\frac{1}{p}} \] is a positively 1-homogeneous convex function on $\mathbb{R}^n$. Our approach also provides a new proof of the original regime $p> 0$. The argument is based on a reduction to a two-dimensional inequality derived from Pr\'ekopa's theorem, which may be of independent interest. As a consequence of this extension, we resolve a nearly 30-year-old question of Richard Gardner and Gaoyong Zhang in the affirmative. In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_p K$ of a convex body $K\subset \mathbb{R}^n$ for $p>-1$. Furthermore, they established that $R_p K$ is convex for $p\geq 0$, but the convexity of $R_p K$ for $p\in (-1,0)$ remained open. We prove that $R_p K$ is convex for all $p>-1$.

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arXiv2026

On the Fourier Mean Bodies of a Convex Body

In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_pK$ of a convex body $K\subset\mathbb R^n$, $p>-1$, which have since become important objects in geometric tomography. In this paper we study the Fourier transforms of the radial functions of $R_pK$. This leads to a new family of star-shaped sets $F_pK$, which we call the Fourier $p$th mean bodies of $K$. We prove Fourier inversion formulas connecting $R_pK$ and $F_pK$, realizing them as $p$-intersection bodies in the sense of A. Koldobsky. We develop the basic affine geometry of $F_pK$; this includes affine invariance and monotonicity properties. We identify the range of $p$ where $F_p K$ is compact in terms of the decay of $|\widehat{\chi_K}|^2$. We show that $F_pK$ is an origin-symmetric convex body for every $0<p\le1$. This range is sharp in general: already for the cube, $F_p[-1,1]^n$ is not convex for $1<p<2$ and $n\geq 2,$ while $F_p[-1,1]^n$ is not compact for $p\geq 2$. We further investigate the features Fourier mean bodies share with intersection bodies: we prove Hensley-type estimates for $F_pK$ when $K$ is isotropic and investigate a few affine isoperimetric inequalities.

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arXiv2025

Affine isoperimetric inequalities for the first eigenvalue of the $m$-th order Affine $p$-Laplace Operator

Recently, Haddad, Jim\'enez, and Montenegro introduced the affine $p$-Laplace operator, $p>1$, and studied associated affine versions of the isoperimetric inequalities for the first eigenvalue of the affine $p$-Laplace operator, including the affine Faber-Krahn inequality and affine Talenti inequality. In this work, we introduce the $m$th-order $p$-Laplace operator $\Delta_{Q,p}^\mathcal{A} f$, which recovers the affine $p$-Laplace operator when $m=1$ and $Q$ is a symmetric interval. Given $n,m \in \mathbb{N}$, a sufficiently smooth convex body $Q \subset \mathbb{R}^m$, a bounded, open set $\Omega \subset \mathbb{R}^n$ and $p >1$, we investigate the eigenvalue problem \[\begin{cases} \Delta_{Q,p}^\mathcal{A} f = \lambda_{1,p}^\mathcal{A}(Q,\Omega) |f|^{p-2} f &\text{ in } \Omega; \\ f=0 & \text{ on } \partial \Omega, \end{cases} \] for $f \in W^{1,p}_0(\Omega)$. Finally, we establish $m$th-order extensions of the affine Talenti inequality and affine Faber-Krahn inequality, which, upon choosing $m=1$, yield new, asymmetric versions of those aforementioned inequalities.

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arXiv2025

Gr\"unbaum's inequality for Gaussian and convex probability measures

A celebrated result in convex geometry is Gr\"unbaum'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 Gr\"unbaum-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-Gr\"unbaum'' 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 ``$s$-Gr\"unbaum'' inequalities can hold only when $s > -1$. However, for convex measures on the real line, we prove Gr\"unbaum-type inequalities involving their cumulative distribution function.

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arXiv2025

Some comments on the mth-order Projection Bodies

The celebrated Petty's projection inequality is a sharp upper bound for the volume of the polar projection body of a convex body. Lutwak introduced the concept of mixed projection bodies and extended Petty's projection inequality. Alonso-Guti\'{e}rrez later did a stability result for Petty's projection inequality. In 1970, Schneider introduced the $m$th-order setting and extended the difference body to that setting. In a previous work, we, working with Haddad, Putterman, Roysdon, and Ye, established an extension of the projection body operator to this setting. In this note, we continue this study for the mixed projection body operator as well as the question of stability.

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arXiv2025

On the polar of Schneider's difference body

In 1970, Schneider introduced the $m$th-order extension of the difference body $DK$ of a convex body $K\subset\mathbb R^n$, the convex body $D^m(K)$ in $\mathbb R^{nm}$. He conjectured that its volume is minimized for ellipsoids when the volume of $K$ is fixed. In this work, we solve a dual version of this problem: we show that the volume of the polar body of $D^m(K)$ is maximized precisely by ellipsoids. For $m=1$ this recovers the symmetric case of the celebrated Blaschke-Santal\'o inequality. We also show that Schneider's conjecture cannot be tackled using standard symmetrization techniques, contrary to this new inequality. As an application for our results, we prove Schneider's conjecture asymptotically \'a la Bourgain-Milman. We also consider a functional version.

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arXiv2025

On Moment-Entropy inequalities in the space of matrices

In a series of works, Lutwak, Yang and Zhang established what could be called affine information theory, which is the study of moment-entropy and Fisher-information-type inequalities that are invariant with respect to affine transformations for random vectors. Their set of tools stemmed from sharp affine isoperimetric inequalities in the $L^p$ Brunn-Minkowski theory of convex geometry they had established. In this work, we generalize the affine information theory to the setting of matrices. These inequalities on the space of $n\times m$ matrices are induced by the interaction between $\mathbb{R}^n$ with its Euclidean structure and $\mathbb{R}^m$ equipped with a pseudo-norm.

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arXiv2024

On a Santal\'o 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-Santal\'o 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.

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arXiv2024

On the $m$th-order Affine P\'olya-Szeg\"o Principle

An affine P\'olya-Szeg\"o principle for a family of affine energies, with equality condition characterization, is demonstrated. In particular, this recovers, as special cases, the $L^p$ affine P\'olya-Szeg\"o principles due to Cianchi, Lutwak, Yang and Zhang, and subsequently Haberl, Schuster and Xiao. Various applications of this new P\'olya-Szeg\"o principle are shown.

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arXiv2024

The Weighted $L^p$ Minkowski Problem

The Minkowski problem in convex geometry concerns showing that a given Borel measure on the unit sphere is, up to perhaps a constant, some type of surface area measure of a convex body. Two types of Minkowski problems in particular are an active area of research: $L^p$ Minkowski problems, introduced by Lutwak and (Lutwak, Yang, and Zhang), and weighted Minkowski problems, introduced by Livshyts. For the latter, the Gaussian Minkowski problem, whose primary investigators were (Huang, Xi and Zhao), is the most prevalent. In this work, we consider weighted surface area in the $L^p$ setting. We propose a framework going beyond the Gaussian setting by focusing on rotationally invariant measures, mirroring the recent development of the Gardner-Zvavitch inequality for rotationally invariant, log-concave measures. Our results include existence for all $p \in \mathbb R$ (with symmetry assumptions in certain instances). We also have uniqueness for $p \geq 1$ under a concavity assumption. Finally, we obtain results in the so-called "small mass regime" using degree theory, as instigated in the Gaussian case by (Huang, Xi and Zhao).

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arXiv2024

On the $m\mathrm{th}$-Order Weighted Projection Body Operator and Related Inequalities

For a convex body $K$ in $\mathbb R^n$, the inequalities of Rogers-Shephard and Zhang, written succinctly, are $$\text{vol}_n(DK)\leq \binom{2n}{n} \text{vol}_n(K) \leq \text{vol}_n(n\text{vol}_n(K)Π^\circ K).$$ Here, $DK=\{x\in\mathbb R^n:K\cap(K+x)\neq \emptyset\}$ is the difference body of $K$, and $Π^\circ K$ is the polar projection body of $K$. There is equality in either if, and only if, $K$ is a $n$-dimensional simplex. In fact, there exists a collection of convex bodies, the so-called radial mean bodies $R_p K$ introduced by Gardner and Zhang, which continuously interpolates between $DK$ and $Π^\circ K$. For $m\in\mathbb N$, Schneider defined the $m$th-order difference body of $K$ as $$D^m(K)=\{(x_1,\dots,x_m)\in\mathbb R^{nm}:K\cap_{i=1}^m(K+x_i)\neq \emptyset\}\subset \mathbb R^{nm}$$ and proved the $m$th-order Rogers-Shephard inequality. In a prequel to this work, the authors, working with Haddad, extended this $m$th-order concept to the radial mean bodies and the polar projection body, establishing the associated Zhang's projection inequality. In this work, we introduce weighted versions of the above-mentioned operators by replacing the Lebesgue measure with measures that have density. The weighted version of these operators in the $m=1$ case was first done by Roysdon (difference body), Langharst-Roysdon-Zvavitch (polar projection body) and Langharst-Putterman (radial mean bodies). This work can be seen as a sequel to all those works, extending them to $m$th-order. In the last section, we extend many of these ideas to the setting of generalized volume, first introduced by Gardner-Hug-Weil-Xing-Ye.

math.FA↗

arXiv2024

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.

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arXiv2024

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.

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arXiv2023

General Higher Order $L^p$ Mean Zonoids

In 1970, Schneider introduced the higher-order difference body and the associated Rogers-Shephard inequality. Recently, Haddad, Langharst, Putterman, Roysdon and Ye expanded the concept to a burgeoning higher-order Brunn-Minkowski theory. In 1991, Zhang introduced mean zonoids of a convex body, which was extended to the Firey-Brunn-Minkowski theory setting by Xi, Guo and Leng in 2014. In this note, we extend these $L^p$ mean zonoids to the higher-order setting and establish the associated isoperimetric inequality.

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arXiv2023

Weighted Minkowski's Existence Theorem and Projection Bodies

The Brunn-Minkowski Theory has seen several generalizations over the past century. Many of the core ideas have been generalized to measures. With the goal of framing these generalizations as a weighted Brunn-Minkowski theory, we prove the Minkowski existence theorem for a large class of Borel measures with continuous density, denoted by $Λ^n$: for $ν$ a finite, even Borel measure on the unit sphere and even $μ\inΛ^n$, there exists a symmetric convex body $K$ such that $$dν(u)=c_{μ,K}dS^μ_{K}(u),$$ where $c_{μ,K}$ is a quantity that depends on $μ$ and $K$ and $dS^μ_{K}(u)$ is the surface area-measure of $K$ with respect to $μ$. Examples of measures in $Λ^n$ are homogeneous measures (with $c_{μ,K}=1$) and probability measures with radially decreasing densities (e.g. the Gaussian measure). We will also consider weighted projection bodies $Π_μK$ by classifying them and studying the isomorphic Shephard problem: if $μ$ and $ν$ are even, homogeneous measures with density and $K$ and $L$ are symmetric convex bodies such that $Π_μ K \subset Π_ν L$, then can one find an optimal quantity $\mathcal{A}>0$ such that $μ(K)\leq \mathcal{A}ν(L)$? Among other things, we show that, in the case where $μ=ν$ and $L$ is a projection body, $\mathcal{A}=1$.

math.FA↗

arXiv2023

Higher-Order Lp Isoperimetric and Sobolev Inequalities

Schneider introduced an inter-dimensional difference body operator on convex bodies and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in $\mathbb R^n$ from those in $\mathbb R^n$, were replaced by inter-dimensional simplicial operators, which generate convex bodies in $\mathbb R^{nm}$ from those in $\mathbb R^{n}$ (or vice versa). In this work, we treat the $L^p$ extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary $m$-dimensional convex bodies containing the origin. We establish $m$th-order $L^p$ isoperimetric inequalities, including the $m$th-order versions of the $L^p$ Petty projection inequality, $L^p$ Busemann-Petty centroid inequality, $L^p$ Santal\'o inequalities, and $L^p$ affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals $(\mathbb R^n, \|\cdot\|_E) \to (\mathbb R^m, \|\cdot\|_F)$.

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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.

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