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Ferenc Fodor

Publications and source records attributed to Ferenc Fodor.

At least 19 recordsLinked to original sources
arXiv2026

Sylvester's four point problem for ball-convex bodies

We investigate Sylvester's classical four-point problem for ball-convex bodies, where convex hulls are replaced by intersections of balls of a fixed radius $R$. We show that the Efron--Buchta identities persist in this framework, so that the distribution of the number of vertices of the $R$-ball-convex hull of $n$ uniform random points is determined by the moments of the volumes of the hulls of its subsamples, and vice versa. For three and four uniform random points in an arbitrary planar ball-convex body, we determine the Sylvester probabilities of the $R$-ball-convex hull. We also give an extension of Groemer's inequality for the expected area of the ball-convex hull of $n$ random points and show that it is minimised by the disc of the same area. In particular, this yields an isoperimetric inequality for the ball-convex analogue of the affine length of the boundary curve. For the unit disc, we can express the Sylvester probabilities using dilogarithmic functions of the radius $R$, and give explicit values at $R=1$. As $R\to\infty$, these recover the classical Sylvester probabilities. As another consequence, we determine the exact distribution function of the circumradius of three uniform random points in he disc on $[1,\infty)$.

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arXiv2026

Optimal stability of P\'al's isominwidth inequality for ball convex bodies in planes of constant curvature

P\'al's isominwidth inequality (1921) answered the Kakeya needle problem (1917) for convex sets. It states that among convex bodies of fixed minimum width $w$ in the Euclidean plane, the regular triangle has minimal area. The isominwidth inequality was generalized to the $2$-dimensional sphere by Bezdek and Blekherman and Freyer and Sagmeister (arXiv:2411.11462). Interestingly, in hyperbolic space, no minimizer exists, as shown by B\"or\"oczky, Freyer and Sagmeister (arXiv:2502.04427). The stability of the Euclidean P\'al inequality with respect to the Hausdorff metric and the symmetric difference metric was proved by Lucardesi and Zucco (arXiv:2405.18294). Fodor, Robock and Sagmeister (arXiv:2602.19300) proved $r$-ball convex analogs of the isominwidth inequality in all three constant curvature planes connecting P\'al's theorem with the Blaschke--Lebesgue inequality. In this paper, we prove optimal stability versions of this statement with respect to the Hausdorff distance and the symmetric difference metric in all three constant curvature planes.

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arXiv2026

Higher moments of intrinsic volumes of random beta-prime polytopes

We consider beta-prime polytopes, i.e., the convex hulls of iid random points chosen according to beta-prime distributions in $\mathbb{R}^d$. After suitable scaling, beta-prime polytopes converge in distribution to the convex hulls of Poisson point processes with power-law intensity functions. We prove moment convergence for the volume and all intrinsic volumes. Beta-prime polytopes are the push-forwards of spherical random polytopes on the upper open half-sphere of the unit sphere $S^d\subset \mathbb{R}^{d+1}$. We prove convergence of moments of the spherical volume difference of the half-sphere and the spherical random polytopes.

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arXiv2026

P\'al's isominwidth inequality for ball convex bodies in planes of constant curvature

P\'al's classical isominwidth inequality states that the regular triangle has minimal area among plane convex bodies of minimal width $w$. A similar result is the Blaschke--Lebesgue inequality that states that Reuleaux triangles minimize the area among bodies of constant width $w$ in the plane. In this paper, we connect these two problems by solving the isominwidth problem for $r$-ball convex bodies in the Euclidean, hyperbolic and spherical planes.

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arXiv2025

Central diagonal sections of Gaussian cubes

The investigation of the volume, surface area, and other geometric properties of sections of convex bodies, and in particular cubes, has a long history and a rich literature. However, much less is known when the cube has a volume distribution that is different from the Lebesgue measure; for example, a Gaussian density. We study the probability densities in the standard cube $B^n_\infty=[-1,1]^n$ of $\mathbb R^n$ generated by $e^{-b\|x\|^2}$, $b> 0$. We prove that the limit of the induced Gaussian-type volume of hyperplane sections of $B^n_\infty$ through the origin and orthogonal to a main diagonal is \[ \sqrt{\frac b\pi}\left (1-4\frac{e^{-b}\sqrt{b}}{2\sqrt{\pi}\mathrm{erf}(\sqrt{b})}\right)^{-\frac12}, \] as $n\to\infty$. This extends the well-known result of Hensley (1979) for the Lebesgue measure and continues the investigations initiated by Barthe, Gu\'edon, Mendelson, Naor (2005), Zvavitch (2008), and K\"onig, Koldobski (2013).

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arXiv2025

On generalized disc-polygons in plane convex bodies with a higher degree of smoothness

We prove power series expansions for the expectations of the number of vertices and missed area of random $L$-convex polygons in planar convex bodies with sufficiently smooth boundaries. Random $L$-convex polygons arise as the intersection of all translates of a fixed convex set $L$ that contain i.i.d. uniform random points from a suitable plane convex body $K$. Our results extend the asymptotic formulas proved in Fodor, Papv\'ari and V\'igh (2020) and Fodor and Montenegro (2024), and have consequences about $L$-convex floating bodies and relative affine surface area that were investigated by Sch\"utt, Werner and Yalikun (2025).

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arXiv2025

Variances and central limit theorems for random beta-polytopes and in other geometric models

We prove matching asymptotic lower and upper bounds on the variances of the intrinsic volumes and the number of $k$-faces of $d$-dimensional random beta-polytopes. Using Stein's methods, we establish central limit theorems for the intrinsic volumes. We also prove asymptotic upper bounds on the variances of the volume and vertex number of spherical random polytopes in spherical convex bodies, and hyperbolic random polytopes in convex bodies in hyperbolic space. Moreover, we consider a circumscribed model on the sphere.

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arXiv2024

Strengthened inequalities for the mean width and the $\ell$-norm of origin symmetric convex bodies

Barthe, Schechtman and Schmuckenschl\"ager proved that the cube maximizes the mean width of symmetric convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball, and the regular crosspolytope minimizes the mean width of symmetric convex bodies whose L\"owner ellipsoid is the Euclidean unit ball. Here we prove close to be optimal stronger stability versions of these results, together with their counterparts about the $\ell$-norm based on Gaussian integrals. We also consider related stability results for the mean width and the $\ell$-norm of the convex hull of the support of even isotropic measures on the unit sphere.

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arXiv2023

A central limit theorem for random disc-polygons in smooth convex discs

In this paper we prove a quantitative central limit theorem for the area of uniform random disc-polygons in smooth convex discs whose boundary is $C^2_+$. We use Stein's method and the asymptotic lower bound for the variance of the area proved by Fodor, Gr\"unfelder and V\'igh (2022).

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arXiv2021

Variance bounds for disc-polygons

We prove asymptotic lower bounds on the variance of the number of vertices and missed area of random disc-polygons in convex discs whose boundary is $C_+^2$ smooth. The established lower bounds are of the same order as the upper bounds proved previously by Fodor and V\'{\i}gh (2018).

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arXiv2021

On random disc-polygons in a disc-polygon

We prove asymptotic formulas for the expectation of the vertex number and missed area of uniform random disc-polygons in convex disc-polygons. Our statements are the $r$-convex analogues of the classical results of R\'enyi and Sulanke (1964) about random polygons in convex polygons.

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arXiv2020

Central diagonal sections of the $n$-cube

We prove that the volume of central hyperplane sections of a unit cube in $\mathbb{R}^n$ orthogonal to a diameter of the cube is a strictly monotonically increasing function of the dimension for $n\geq 3$. Our argument uses an integral formula that goes back to P\'olya \cite{P} (see also \cite{H} and \cite{B86}) for the volume of central sections of the cube, and Laplace's method to estimate the asymptotic behaviour of the integral. First we show that monotonicity holds starting from some specific $n_0$. Then, using interval arithmetic (IA) and automatic differentiation (AD), we compute an explicit bound for $n_0$, and check the remaining cases between $3$ and $n_0$ by direct computation.

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arXiv2020

Strengthened inequalities for the mean width and the $\ell$-norm

Barthe proved that the regular simplex maximizes the mean width of convex bodies whose John ellipsoid (maximal volume ellipsoid contained in the body) is the Euclidean unit ball; or equivalently, the regular simplex maximizes the $\ell$-norm of convex bodies whose L\"owner ellipsoid (minimal volume ellipsoid containing the body) is the Euclidean unit ball. Schmuckenschl\"ager verified the reverse statement; namely, the regular simplex minimizes the mean width of convex bodies whose L\"owner ellipsoid is the Euclidean unit ball. Here we prove stronger stability versions of these results. We also consider related stability results for the mean width and the $\ell$-norm of the convex hull of the support of centered isotropic measures on the unit sphere.

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arXiv2019

On random approximations by generalized disc-polygons

For two convex discs $K$ and $L$, we say that $K$ is $L$-convex if it is equal to the intersection of all translates of $L$ that contain $K$. In $L$-convexity the set $L$ plays a similar role as closed half-spaces do in the classical notion of convexity. We study the following probability model: Let $K$ and $L$ be $C^2_+$ smooth convex discs such that $K$ is $L$-convex. Select $n$ i.i.d. uniform random points $x_1,\ldots, x_n$ from $K$, and consider the intersection $K_{(n)}$ of all translates of $L$ that contain all of $x_1,\ldots, x_n$. The set $K_{(n)}$ is a random $L$-convex polygon in $K$. We study the expectation of the number of vertices $f_0(K_{(n)})$ and the missed area $A(K\setminus K_{n})$ as $n$ tends to infinity. We consider two special cases of the model. In the first case we assume that the maximum of the curvature of the boundary of $L$ is strictly less than $1$ and the minimum of the curvature of $K$ is larger than $1$. In this setting the expected number of vertices and missed area behave in a similar way as in the classical convex case and in the $r$-spindle convex case (when $L$ is a radius $r$ circular disc). The other case we study is when $K=L$. This setting is special in the sense that an interesting phenomenon occurs: the expected number of vertices tends to a finite limit depending only on $L$. This was previously observed in the special case when $L$ is a circle of radius $r$ (Fodor, Kevei and V\'igh (2014)). We also determine the extrema of the limit of the expectation of the number of vertices of $L_{(n)}$ if $L$ is a convex discs of constant width $1$. The formulas we prove can be considered as generalizations of the corresponding $r$-spindle convex statements proved by Fodor, Kevei and V\'igh (2014).

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arXiv2019

Random ball-polytopes in smooth convex bodies

We study approximations of smooth convex bodies by random ball-polytopes. We examine the following probability model: let $K\subset{\bf R}^d$ be a convex body such that $K$ slides freely in a ball of radius $R>0$ and has $C^2$ smooth boundary. Let $x_1,\ldots, x_n$ be i.i.d. uniform random points in $K$. For $r\geq R$, let $K^r_{(n)}$ denote the intersection of all radius $r$ closed balls that contain $x_1,\ldots, x_n$. Then $K^r_{(n)}$ is a (uniform) random ball-polytope (of radius $r$) in $K$. We study the asymptotic properties of the expectation of the number of facets of $K_{(n)}^r$ as $n\to\infty$. While sufficiently round convex bodies behave in a similar way with respect to random approximation by ball-polytopes as to classical polytopes, an interesting phenomenon can be observed when a unit ball is approximated by unit radius random ball-polytopes: the expected number of facets approaches a finite limit as $n\to\infty$.

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arXiv2019

Perimeter approximation of convex discs in the hyperbolic plane and on the sphere

Eggleston (1957) proved that in the Euclidean plane the best approximating convex $n$-gon to a convex disc $K$ is always inscribed in $K$ if we measure the distance by perimeter deviation. We prove that the analogue of Eggleston's statement holds in the hyperbolic plane, and we give an example showing that it fails on the sphere.

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