arXiv subjects

J. Haddad

Publications and source records attributed to J. Haddad.

8 recordsLinked to original sources

arXiv2026

An approximate counterexample to the Barker--Larman problem in dimension $4$

The Barker--Larman problem asks if a convex body $K \subseteq \mathbb R^n$ containing the Euclidean ball $\mathbb B_n$, such that all the sections of $K$ by hyperplanes tangent to $\mathbb B_n$ have constant $(n-1)$-dimensional volume, must necessarily be a Euclidean ball. In this paper we show a result pointing to a negative answer in dimension $4$. Taking $\lambda_0 = 4 \sqrt{3}\pi$ and any $N \in \mathbb N$, we obtain the existence of a family of convex bodies $K_{\lambda,N}$ with $\lambda \in (\lambda_0-r_N, \lambda_0 + r_N)$, such that the sections of $K_{\lambda,N}$ by hyperplanes tangent to the Euclidean ball, have area within $c |\lambda - \lambda_0|^{\frac{N+1}2}$ of $\lambda$, while the difference between outradius and inradius of $K_{\lambda,N}$ is larger than $C |\lambda - \lambda_0|$. The bodies $K_{\lambda,N}$ are constructed via radial functions as \[\rho_{K_{\lambda,N}}(t) = \cos\left( \sum_{n=0}^N \frac{(\lambda-\lambda_0)^n}{n!} \varphi_n(t) \right)^{-1},\] where $t \in [0,2\pi), \lambda \in \mathbb R$ and $\varphi_n$ are trigonometric polynomials that can be computed explicitly. The convergence of the inner power series when $N \to \infty$ (which is left open) would imply a negative answer to the Barker--Larman problem in dimension $4$. As an example we obtain a convex body whose outradius and inradius differ by more than $0.176$, and the area of the sections oscillate by less than $3 \times 10^{-7}$.

math.MG↗

arXiv2026

On convex bodies with constant non-central sections

We prove that if $C$ is a symmetric convex body of revolution in $\mathbb R^4$ containing the unit Euclidean ball $\mathbb B_4$, such that the sections of $C$ by hyperplanes tangent to $\mathbb B_4$ have constant area $A>0$, then $C$ is a Euclidean ball, provided $\frac 1{\pi} \arctan((\frac{3A}{4\pi})^{1/3})$ satisfies certain arithmetic properties that can be read from its expansion as a continued fraction. We show that the set of values $A$ satisfying these properties has positive Hausdorff dimension.

math.MG↗

arXiv2025

New criteria on positive-definite distributions

We establish several sufficient conditions under which a locally integrable function $f:\mathbb R^n \to \mathbb R$ represents a positive-definite distribution. In particular we consider functions of the form $f(\|x\|)$ where $\|\cdot\|$ is a fixed norm in $\mathbb R^n$.

math.FA↗

arXiv2024

Planar radial mean bodies are convex

The radial mean bodies of parameter $p>-1$ of a convex body $K \subseteq \mathbb R^n$ are radial sets introduced in [4] by Gardner and Zhang. They are known to be convex for $p\geq 0$. We prove that if $K \subseteq \mathbb R^2$ is a convex body, then its radial mean body of parameter $p$ is convex for every $p \in (-1,0)$.

math.MG↗

arXiv2024

The lower dimensional slicing inequality for functions and related distance inequalities

It was shown in [11] that for every origin-symmetric star body $K \subseteq \mathbb R^n$ of volume $1$, every even continuous probability density $f$ on $K$ and $1 \leq k \leq n-1$, there exists a subspace $F \subseteq \mathbb R^n$ of codimension $k$ such that \[ \int_{K \cap F} f \geq c^k (d_{\rm ovr}(K, \mathcal{BP}_k^n))^{-k} \] where $d_{\rm ovr}(K, \mathcal{BP}_k^n)$ is the outer volume ratio distance from $K$ to the class of generalized $k$-intersection bodies, and $c>0$ is a universal constant. The upper bound $d_{\rm ovr}(K, \mathcal{BP}_k^n) \leq c' \sqrt{n/k} \left(\log\left(\frac{en}k\right)\right)^{3/2}$ was established in [13] for every origin-symmetric convex body $K$. In this note we show that there exist an origin-symmetric convex body $K$ of volume $1$ and an even continuous probability density $f$ supported on $K$ such that for every subspace $F$ of codimension $k$, \[ \int_{K \cap F} f \leq \left( c \sqrt{\frac n{k \log(n)} } \right)^{-k}. \] As a consequence we obtain a lower bound for $d_{\rm ovr}(K, \mathcal{BP}_k^n)$ with $K$ a convex body, complementing the upper bound in \cite{koldobsky2011isomorphic}. This is \[c \sqrt{n/k} (\log(n))^{-1/2} \leq \sup_K d_{\rm ovr}(K, \mathcal{BP}_k^n) \leq c' \sqrt{n/k} \left(\log\left(\frac{en}k\right)\right)^{3/2}.\] The case $k=1$ was obtained previously in [5,6].

math.MG↗

arXiv2024

On the volume of convolution bodies in the plane

For every convex body $K \subset \mathbb R^n$ and $\delta \in (0,1)$, the $\delta$-convolution body of $K$ is the set of $x \in \mathbb R^n$ for which $\left|K \cap (K+x)\right|_n \geq \delta \left|K\right|_n$. We show that for $n=2$ and any $\delta \in (0,1)$, ellipsoids do not maximize the volume of the $\delta$-convolution body of $K$, when $K$ runs over all convex bodies of a fixed volume. This behavior is somehow unexpected and contradicts the limit case $\delta \to 1^-$, which is governed by the Petty projection inequality.

math.MG↗

arXiv2021

On explicit representations of isotropic measures in John and L\"owner positions

Given a convex body $K \subseteq \mathbb R^n$ in L\"owner position we study the problem of constructing a non-negative centered isotropic measure supported in the contact points, whose existence is guaranteed by John's Theorem. The method we propose requires the minimization of a convex function defined in an $\frac {n(n+3)}2$ dimensional vector space. We find a geometric interpretation of the minimizer as $\left. \frac{\partial}{\partial r}(A_r, v_r)\right|_{r=1}$, where $A_r K + v_r$ is a one-parameter family of positions of $K$ that are in some sense related to the maximal intersection position of radius $r$ defined recently by Artstein-Avidan and Katzin.

math.MG↗

arXiv2021

Stability of Hypersurfaces in Minkowsky Normed Spaces

We extend to Minkowski spaces the classical result of Barbosa and do Carmo [1] that characterizes the euclidean sphere as the unique compact stable CMC hypersurface of $\mathbb R^n$. More precisely, if $K$ is a smooth convex body in $\mathbb R^n$ with positive Gauss curvature, containing the origin in its interior and $M$ is an immersed hypersurface, there are well defined concepts of surface area measure, normal vector field and principal curvatures of $M$ , with respect to $K$. Thus, we introduce the concept of stability with respect to normal variations and compute the formula of second variation with respect to $K$. Finally we show that if $M$ is compact, has constant mean Minkowski curvature and is stable (with respect to $K$) then $M$ is homothetic to $\partial K$.

math.DG↗