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Michael Roysdon

Publications and source records attributed to Michael Roysdon.

17 recordsLinked to original sources
arXiv2026

On cost-induced Santaló-type inequalities in Polish measure spaces

We introduce a framework for establishing Blaschke-Santaló-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 Santaló-type. We call these equivalent inequalities cost-Santaló inequalities. This definition expands and includes previous notions in the literature. We apply this principle to deduce several new versions of functional Santaló inequalities, including on the space of rectangular matrices and a functional sine Santaló 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 Santaló 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-Santaló inequality, which we explicitly instantiate for matrix spaces.

math.FA↗
arXiv2026

From simplex slicing to sharp reverse Hölder inequalities

Simplex slicing (Webb, 1996) is a sharp upper bound on the volume of central hyperplane sections of the regular simplex. We extend this to sharp bounds in the probabilistic framework of negative moments, and beyond, of centred log-concave random variables, establishing a curious phase transition of the extremising distribution for new sharp reverse Hölder-type inequalities.

math.MG↗
arXiv2025

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

Recently, Haddad, Jiménez, 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 $Δ_{Q,p}^\mathcal{A} f$, 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 $Ω\subset \mathbb{R}^n$ and $p >1$, we investigate the eigenvalue problem \[\begin{cases} Δ_{Q,p}^\mathcal{A} f = λ_{1,p}^\mathcal{A}(Q,Ω) |f|^{p-2} f &\text{ in } Ω; \\ f=0 & \text{ on } \partial Ω, \end{cases} \] for $f \in W^{1,p}_0(Ω)$. 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.

math.FA↗
arXiv2025

Functional Liftings of Restricted Geometric Inequalities

We investigate what we term "generalized sup-convolutions". We show that functional inequalities that enjoy an interpretation as sup-convolution inequalities can be deduced from the special case of indicator functions corresponding to a geometric inequality. As consequences we derive a Borell-Brascamp Lieb inequality for the Gaussian Brunn-Minkowski inequality and give a functional analog of the log-Brunn Minkowski conjecture. Though we focus on Euclidean applications, our results are general and can be directly applied in more abstract settings, like groups or even topological measure spaces without algebraic structure, we instantiate this claim with a Borell-Brascamp-Lieb type inequality for nilpotent Lie groups.

math.FA↗
arXiv2025

Affine Isoperimetric Inequalities for Higher-Order Projection and Centroid Bodies

In 1970, Schneider introduced the $m$th order difference body of a convex body, and also established the $m$th-order Rogers-Shephard inequality. In this paper, we extend this idea to the projection body, centroid body, and radial mean bodies, as well as prove the associated inequalities (analogues of Zhang's projection inequality, Petty's projection inequality, the Busemann-Petty centroid inequality and Busemann's random simplex inequality). We also establish a new proof of Schneider's $m$th-order Rogers-Shephard inequality. As an application, a $m$th-order affine Sobolev inequality for functions of bounded variation is provided.

math.FA↗
arXiv2025

On maximal intersection position for logarithmically concave functions and measures

A new position is introduced and studied for the convolution of log-concave functions, which may be regarded as a functional analogue of the maximum intersection position of convex bodies introduced and studied by Artstein-Avidan and Katzin (2018) and Artstein-Avidan and Putterman (2022). Our main result is a John-type theorem for the maximal intersection position of a pair of log-concave functions, including the corresponding decomposition of the identity. The main result holds under very weak assumptions on the functions; in particular, the functions considered may both have unbounded supports. As an application of our results, we introduce a John-type position for even $\log$-concave measures.

math.FA↗
arXiv2025

A Busemann-Petty Type Problem for Dual Radon Transforms

Inspired by resolution of the Busemann-Petty problem (1956), we consider the following comparison problem for dual Radon transforms: Given a pair of continuous functions defined on the affine Grassmannian whose dual Radon transforms satisfy a pointwise inequality, can their $L^p$ norms be compared in a meaningful way? We characterize the solution to this problem for each $p\geq 1$, and as a consequence of our investigation, we prove reverse $L^p$-$L^q$-estimates for dual Radon transforms. In particular, we reverse an inequality of Solmon (1979).

math.FA↗
arXiv2024

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, $L^p$ Santaló inequalities, and $L^p$ affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals $(\mathbb R^n, \|\cdot\|_E) \to (\mathbb R^m, \|\cdot\|_F)$.

math.MG↗
arXiv2024

On the $m$th-order Affine Pólya-Szegö Principle

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

math.FA↗
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↗
arXiv2023

Comparison Problems for Radon Transforms

Given two non-negative functions $f$ and $g$ such that the Radon transform of $f$ is pointwise smaller than the Radon transform of $g$, does it follow that the $L^p$-norm of $f$ is smaller than the $L^p$-norm of $g$ for a given $p>0$? We consider this problem for the classical and spherical Radon transforms. In both cases we point out classes of functions for which the answer is affirmative, and show that in general the answer is negative if the functions do not belong to these classes. The results are in the spirit of the solution of the Busemann-Petty problem from convex geometry, and the classes of functions that we introduce generalize the class of intersection bodies introduced by Lutwak in 1988. We also deduce slicing inequalities that are related to the well-known Oberlin-Stein type estimates for the Radon transform.

math.FA↗
arXiv2022

General Measure Extensions of Projection Bodies

The inequalities of Petty and Zhang are affine isoperimetric-type inequalities providing sharp bounds for $\text{vol}^{n-1}_{n}(K)\text{vol}_n(Π^\circ K),$ where $ΠK$ is a projection body of a convex body $K$. In this paper, we present a number of generalizations of Zhang's inequality to the setting of arbitrary measures. In addition, we introduce extensions of the projection body operator $Π$ to the setting of arbitrary measures and functions, while providing associated inequalities for this operator; in particular, Zhang-type inequalities. Throughout, we apply shown results to the standard Gaussian measure.

math.FA↗
arXiv2022

On Multiple $L_p$-curvilinear-Brunn-Minkowski inequalities

We construct the extension of the curvilinear summation for bounded Borel measurable sets to the $L_p$ space for multiple power parameter $\barα=(α_1, \cdots, α_{n+1})$ when $p>0$. Based on this $L_{p,\barα}$-curvilinear summation for sets and concept of {\it compression} of sets, the $L_{p,\barα}$-curvilinear-Brunn-Minkowski inequality for bounded Borel measurable sets and its normalized version are established. Furthermore, by utilizing the hypo-graphs for functions, we enact a brand new proof of $L_{p,\barα}$ Borell-Brascamp-Lieb inequality, as well as its normalized version, for functions containing the special case of $L_{p}$ Borell-Brascamp-Lieb inequality through the $L_{p,\barα}$-curvilinear-Brunn-Minkowski inequality for sets. Moreover, we propose the multiple power $L_{p,\barα}$-supremal-convolution for two functions together with its properties. Last but not least, we introduce the definition of the surface area originated from the variation formula of measure in terms of the $L_{p,\barα}$-curvilinear summation for sets as well as $L_{p,\barα}$-supremal-convolution for functions together with their corresponding Minkowski type inequalities and isoperimetric inequalities for $p\geq1,$ etc.

math.FA↗
arXiv2020

On $L_p$-Brunn-Minkowski type and $L_p$-isoperimetric type inequalities for general measures

In 2011 Lutwak, Yang and Zhang extended the definition of the $L_p$-Minkowski convex combination ($p \geq 1$) introduced by Firey in the 1960s from convex bodies containing the origin in their interiors to all measurable subsets in $\mathbb{R}^n$, and as a consequence, extended the $L_p$-Brunn-Minkowski inequality ($L_p$-BMI) to the setting of all measurable sets. In this paper, we present a functional extension of their $L_p$-Minkowski convex combination---the $L_{p,s}$--supremal convolution and prove the $L_p$-Borell-Brascamp-Lieb type ($L_p$-BBL) inequalities. Based on the $L_p$-BBL type inequalities for functions, we extend the $L_p$-BMI for measurable sets to the class of Borel measures on $\mathbb{R}^n$ having $\left(\frac{1}{s}\right)$-concave densities, with $s \geq 0$; that is, we show that, for any pair of Borel sets $A,B \subset \mathbb{R}^n$, any $t \in [0,1]$ and $p\geq 1$, one has \[ μ((1-t) \cdot_p A +_p t \cdot_p B)^{\frac{p}{n+s}} \geq (1-t) μ(A)^{\frac{p}{n+s}} + t μ(B)^{\frac{p}{n+s}}, \] where $μ$ is a measure on $\mathbb{R}^n$ having a $\left(\frac{1}{s}\right)$-concave density for $0 \leq s < \infty$. Additionally, with the new defined $L_{p,s}$--supremal convolution for functions, we prove $L_p$-BMI for product measures with quasi-concave densities and for log-concave densities, $L_p$-Prékopa-Leindler type inequality ($L_p$-PLI) for product measures with quasi-concave densities, $L_p$-Minkowski's first inequality ($L_p$-MFI) and $L_p$ isoperimetric inequalities ($L_p$-ISMI) for general measures, etc. Finally a functional counterpart of the Gardner-Zvavitch conjecture is presented for the $p$-generalization.

math.FA↗
arXiv2019

Rogers-Shepard Type Inequalities for Sections

In this paper we address the following question: given a measure $μ$ on $\mathbb{R}^n$, does there exists a constant $C>0$ such that, for any $m$-dimensional subspace $H \subset \mathbb{R}^n$ and any convex body $K \subset \mathbb{R}^n$, the following sectional Rogers-Shephard type inequality holds: \[ μ((K-K) \cap H) \leq C \sup_{y \in \mathbb{R}^n} μ(K \cap (H+y))? \] We show that this inequality is affirmative in the class of measures with radially decreasing densities with the constant $C(n,m) = \binom{n+m}{m}$. We also prove marginal inequalities of the Rogers-Shephard type for $\left(\frac{1}{s}\right)$-concave, $0 \leq s < \infty$, and logarithmically concave functions.

math.MG↗
arXiv2018

On Rogers-Shephard type inequalities for general measures

In this paper we prove a series of Rogers-Shephard type inequalities for convex bodies when dealing with measures on the Euclidean space with either radially decreasing densities, or quasi-concave densities attaining their maximum at the origin. Functional versions of classical Rogers-Shephard inequalities are also derived as consequences of our approach.

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