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Christos Saroglou

Publications and source records attributed to Christos Saroglou.

At least 19 recordsLinked to original sources
arXiv2026

On Ball's conjectured Santaló type inequality

We prove that if $K$ is a symmetric and isotropic convex body in $\mathbb{R}^n$, then $$\int_K\langle x,u\rangle^2\,dx\int_{K^\circ}\langle x,u\rangle^2\,dx\leq \left(\int_{B_2^n}\langle x,u\rangle^2\,dx\right)^2,\qquad\forall u\in\mathbb{R}^n,$$with equality for some $u\neq o$, if and only if $K$ is a Euclidean ball. This confirms a conjecture by Keith Ball (1986), stating that for any symmetric convex body $K$ in $\mathbb{R}^n$, it holds $$\int_K\int_{K^\circ}\langle x,y\rangle^2\,dx\,dy\leq \int_{B_2^n}\int_{B_2^n}\langle x,y\rangle^2\,dx\,dy,$$with equality if and only if $K$ is an ellipsoid. Fortunately, our method for proving Ball's conjectured inequality admits a quantitative stability refinement, which in turn yields an asymptotically optimal stability version of the Blaschke-Santaló inequality for origin symmetric convex bodies in terms of the symmetric difference metric. This resolves another well known open problem.

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arXiv2026

A note on the $L_p$-Brunn-Minkowski inequality for intrinsic volumes and the $L_p$-Christoffel-Minkowski problem

The first goal of this paper is to improve some of the results in \cite{BCPR}. Namely, we establish the $L_p$-Brunn-Minkwoski inequality for intrinsic volumes for origin-symmetric convex bodies that are close to the ball in the $C^2$ sense for a certain range of $p<1$ (including negative values) and we prove that this inequality does not hold true in the entire class of origin-symmetric convex bodies for any $p<1$. The second goal is to establish a uniqueness result for the (closely related) $L_p$-Christoffel-Minkowski problem. More specifically, we show uniqueness in the symmetric case when $p\in[0,1)$ and the data function $g$ in the right hand side is sufficiently close to the constant 1. One of the main ingredients of the proof is the existence of upper and lower bounds for the (convex) solution, that depend only $\|\log g\|_{L^\infty}$, a fact that might be of independent interest.

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arXiv2025

Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$

We prove the $C^0$ estimate for the $L_p$ $q$th dual Minkowski problem on $S^2$ under fairly general conditions; namely, when $p$ lies in [0,1) and $q>2+p$, and the $L_p$ $q$th dual curvarture is bounded and bounded away from zero. We note that it is known that the analogous $C^0$ estimate does not hold if $p<-1$ and $q=3$. As a corollary of our $C^0$ estimate, we deduce the uniqueness of the solution of the near isotropic $q$th $L_p$ dual Minkowski problem on $S^2$ if $q$ is close to 3 and the $q$th $L_p$ dual curvature is Holder close to be the constant one function.

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arXiv2024

Complex and Quaternionic Analogues of Busemann's Random Simplex and Intersection Inequalities

In this paper, we extend two celebrated inequalities by Busemann -- the random simplex inequality and the intersection inequality -- to both complex and quaternionic vector spaces. Our proof leverages a monotonicity property under symmetrization with respect to complex or quaternionic hyperplanes. Notably, we demonstrate that the standard Steiner symmetrization, contrary to assertions in a paper by Grinberg, does not exhibit this monotonicity property.

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arXiv2024

Uniqueness when the $L_p$ curvature is close to be a constant for $p\in[0,1)$

For fixed positive integer $n$, $p\in[0,1]$, $a\in(0,1)$, we prove that if a function $g:\mathbb{S}^{n-1}\to \mathbb{R}$ is sufficiently close to 1, in the $C^a$ sense, then there exists a unique convex body $K$ whose $L_p$ curvature function equals $g$. This was previously established for $n=3$, $p=0$ by Chen, Feng, Liu \cite{CFL22} and in the symmetric case by Chen, Huang, Li, Liu \cite{CHLL20}. Related, we show that if $p=0$ and $n=4$ or $n\leq 3$ and $p\in[0,1)$, and the $L_p$ curvature function $g$ of a (sufficiently regular, containing the origin) convex body $K$ satisfies $λ^{-1}\leq g\leq λ$, for some $λ>1$, then $\max_{x\in\mathbb{S}^{n-1}}h_K(x)\leq C(p,λ)$, for some constant $C(p,λ)>0$ that depends only on $p$ and $λ$. This also extends a result from Chen, Feng, Liu \cite{CFL22}. Along the way, we obtain a result, that might be of independent interest, concerning the question of when the support of the $L_p$ surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the $L_p$-Minkowksi problem, for $-n<p<0$.

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arXiv2022

On a $j$-Santaló Conjecture

Let $k\geq 2$ be an integer. In the spirit of Kolesnikov-Werner \cite{KW}, for each $j\in\{2,\ldots,k\}$, we conjecture a sharp Santaló type inequality (we call it $j$-Santaló conjecture) for many sets (or more generally for many functions), which we are able to confirm in some cases, including the case $j=k$ and the unconditional case. Interestingly, the extremals of this family of inequalities are tuples of the $l_j^n$-ball. Our results also strengthen one of the main results in \cite{KW}, which corresponds to the case $j=2$. All members of the family of our conjectured inequalities can be interpreted as generalizations of the classical Blaschke-Santaló inequality. Related, we discuss an analogue of a conjecture due to K. Ball \cite{Ball-conjecture} in the multi-entry setting and establish a connection to the $j$-Santaló conjecture.

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arXiv2021

A non-existence result for the $L_p$-Minkowski problem

We show that given a real number $p<1$, a positive integer $n$ and a proper subspace $H$ of $\mathbb{R}^n$, the measure on the Euclidean sphere $\mathbb{S}^{n-1}$, which is concentrated in $H$ and whose restriction to the class of Borel subsets of $\mathbb{S}^{n-1}\cap H$ equals the spherical Lebesgue measure on $\mathbb{S}^{n-1}\cap H$, is not the $L_p$-surface area measure of any convex body. This, in particular, disproves a conjecture from [Bianchi, Böröczky, Colesanti, Yang, The $L_p$-Minkowski problem for $-n<p<1$, Adv. Math. (2019)].

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arXiv2020

On a non-homogeneous version of a problem of Firey

We investigate the uniqueness for the Monge-Ampère type equation \begin{equation} \label{eq-abstract} det(u_{ij}+δ_{ij}u)_{i,j=1}^{n-1}=G(u),\ \ \ \ \ \ \ (*)\end{equation}on $S^{n-1}$, where $u$ is the restriction of the support function on the sphere $S^{n-1}$ of a convex body that contains the origin in its interior and $G:(0,\infty)\to(0,\infty)$ is a continuous function. The problem was initiated by Firey (1974) who, in the case $G(θ)=θ^{-1}$, asked if $u\equiv 1$ is the unique solution to (*). Recently, Brendle, Choi and Daskalopoulos $[9]$ proved that if $G(θ)=θ^{-p}$, $p>-n-1$, then $u$ has to be constant, providing in particular a complete solution to Firey's problem. Our primary goal is to obtain uniqueness (or nearly uniqueness) results for (*) for a broader family of functions $G$. Our approach is very different than the techniques developed in $[9]$.

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arXiv2019

Functions with isotropic sections

We prove a local version of a recently established theorem by Myroshnychenko, Ryabogin and the second named author. More specifically, we show that if $n\geq 3$, $g:\mathbb{S}^{n-1}\to\mathbb{R}$ is an even bounded measurable function, $U$ is an open subset of $\mathbb{S}^{n-1}$ and the restriction (section) of $f$ onto any great sphere perpendicular to $U$ is isotropic, then ${\cal C}(g)|_U=c+\langle a,\cdot\rangle$ and ${\cal R}(g)|_U=c'$, for some fixed constants $c,c'\in\mathbb{R}$ and for some fixed vector $a\in \mathbb{R}^n$. Here, ${\cal C}(g)$ denotes the cosine transform and ${\cal R}(g)$ denotes the Funk transform of $g$. However, we show that $g$ does not need to be equal to a constant almost everywhere in $U^\perp:=\bigcup_{u\in U}(\mathbb{S}^{n-1}\cap u^\perp)$. For the needs of our proofs, we obtain a new generalization of a result from classical differential geometry, in the setting of convex hypersurfaces, that we believe is of independent interest.

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arXiv2018

On some problems concerning symmetrization operators

In [G. Bianchi, R. J. Gardner and P. Gronchi, Symmetrization in Geometry, Adv. Math., vol. 306 (2017), 51-88], a systematic study of symmetrization operators on convex sets and their properties is conducted. In the end of their article, the authors pose several open questions. The primary goal of this manuscript is to study these questions.

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arXiv2018

Wulff shapes and a characterization of simplices via a Bezout type inequality

Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii we study the following Bezout type inequality for mixed volumes $$ V(L_1,\dots,L_{n})V_n(K)\leq V(L_1,K[{n-1}])V(L_2,\dots, L_{n},K). $$ We show that the above inequality characterizes simplices, i.e. if $K$ is a convex body satisfying the inequality for all convex bodies $L_1, \dots, L_n \subset {\mathbb R}^n$, then $K$ must be an $n$-dimensional simplex. The main idea of the proof is to study perturbations given by Wulff shapes. In particular, we prove a new theorem on differentiability of the support function of the Wulff shape, which is of independent interest. In addition, we study the Bezout inequality for mixed volumes introduced in arXiv:1507.00765 . We introduce the class of weakly decomposable convex bodies which is strictly larger than the set of all polytopes that are non-simplices. We show that the Bezout inequality in arXiv:1507.00765 characterizes weakly indecomposable convex bodies.

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arXiv2017

Estimating volume and surface area of a convex body via its projections or sections

The main goal of this paper is to present a series of inequalities connecting the surface area measure of a convex body and surface area measure of its projections and sections. We present a solution of a question from S. Campi, P. Gritzmann and P. Gronchi regarding the asymptotic behavior of the best constant in a recently proposed reverse Loomis-Whitney inequality. Next we give a new sufficient condition for the slicing problem to have an affirmative answer, in terms of the least "outer volume ratio distance" from the class of intersection bodies of projections of at least proportional dimension of convex bodies. Finally, we show that certain geometric quantities such as the volume ratio and minimal surface area (after a suitable normalization) are not necessarily close to each other.

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arXiv2016

Star bodies with completely symmetric sections

We say that a star body $K$ is completely symmetric if it has centroid at the origin and its symmetry group $G$ forces any ellipsoid whose symmetry group contains $G$, to be a ball. In this short note, we prove that if all central sections of a star body $L$ are completely symmetric, then $L$ has to be a ball. A special case of our result states that if all sections of $L$ are origin symmetric and 1-symmetric, then $L$ has to be a Euclidean ball. This answers a question from \cite{R2}. Our result is a consequence of a general theorem that we establish, stating that if the restrictions in almost all equators of a real function $f$ defined on the sphere, are isotropic functions, then $f$ is constant a.e. In the last section of this note, applications, improvements and related open problems are discussed and two additional open questions from \cite{R} and \cite{R2} are answered.}

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arXiv2015

Characterization of Simplices via the Bezout Inequality for Mixed volumes

We consider the following Bezout inequality for mixed volumes: $$V(K_1,\dots,K_r,Δ[{n-r}])V_n(Δ)^{r-1}\leq \prod_{i=1}^r V(K_i,Δ[{n-1}])\ \text{ for }2\leq r\leq n.$$ It was shown previously that the inequality is true for any $n$-dimensional simplex $Δ$ and any convex bodies $K_1, \dots, K_r$ in $\mathbb{R}^n$. It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies $K_1, \dots, K_r$ in $\mathbb{R}^n$. In this paper we prove that this is indeed the case if we assume that $Δ$ is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex $n$-polytopes. In addition, we show that if a body $Δ$ satisfies the Bezout inequality for all bodies $K_1, \dots, K_r$ then the boundary of $Δ$ cannot have strict points. In particular, it cannot have points with positive Gaussian curvature.

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arXiv2015

Iterations of the projection body operator and a remark on Petty's conjectured projection inequality

We prove that if a convex body has absolutely continuous surface area measure, whose density is sufficiently close to the constant, then the sequence $\{Π^mK\}$ of convex bodies converges to the ball with respect to the Banach-Mazur distance, as $m\rightarrow\infty$. Here, $Π$ denotes the projection body operator. Our result allows us to show that the ellipsoid is a local solution to the conjectured inequality of Petty and to improve a related inequality of Lutwak.

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arXiv2014

On the equivalence between two problems of asymmetry on convex bodies

The simplex was conjectured to be the extremal convex body for the two following "problems of asymmetry":\\ P1) What is the minimal possible value of the quantity $\max_{K'} |K'|/|K|$? Here, $K'$ ranges over all symmetric convex bodies contained in $K$.\\ P2) What is the maximal possible volume of the Blaschke-body of a convex body of volume 1?\\ Our main result states that (P1) and (P2) admit precisely the same solutions. This complements a result from [{\rm K. Böröczky, I. Bárány, E. Makai Jr. and J. Pach}, Maximal volume enclosed by plates and proof of the chessboard conjecture], Discrete Math. {\bf 69} (1986), 101--120], stating that if the simplex solves (P1) then the simplex solves (P2) as well.

math.FA↗
arXiv2014

More on logarithmic sums of convex bodies

We prove that the log-Brunn-Minkowski inequality (log-BMI) for the Lebesque measure in dimension $n$ would imply the log-BMI and, therefore, the B-conjecture for any log-concave density in dimension $n$. As a consequence, we prove the log-BMI and the B-conjecture for any log-concave density, in the plane. Moreover, we prove that the log-BMI reduces to the following: For each dimension $n$, there is a density $f_n$, which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density $f_n$. As byproduct of our methods, we study possible log-concavity of the function $t\mapsto |(K+_p\cdot e^tL)^{\circ}|$, where $p\geq 1$ and $K$, $L$ are symmetric convex bodies, which we are able to prove in some instances and as a further application, we confirm the variance conjecture in a special class of convex bodies. Finally, we establish a non-trivial dual form of the log-BMI.

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