arXiv subjects

Márton Naszódi

Publications and source records attributed to Márton Naszódi.

At least 19 recordsLinked to original sources

arXiv2026

The VC dimension of partial concept classes via Radon's theorem

Following Alon, Hanneke, Holzman, and Moran (FOCS 2021), we define a partial concept class (PCC) as a family of partial functions $f: V\to\{0,1,\ast\}$; equivalently, its concepts partition the ground set into black ($f^{-1}(1)$), grey ($f^{-1}(\ast)$), and white parts ($f^{-1}(0)$). Its VC dimension is defined by shattering sets on which the value $\ast$ is not taken. We study two geometric PCCs in real Banach spaces, both with a margin $δ>0$: expanded half-spaces, where the grey part is a strip of width at least $δ$ adjacent to a half-space, and expanded balls, where the grey part is an annulus of width $δ$ around a unit radius ball. Our main results are dimension-free upper bounds on the VC dimension of the PCC of expanded balls in $L_p\parenthμ$, $1\le p<\infty$, including the non-Euclidean and algorithmically particularly relevant case $\ell^d_1$. These bounds depend on the margin and on the radii, but not on the ambient dimension or the underlying measure space. These are extensions of the work of Bourneuf, Charbit, and Thomassé (FOCS 2025) who studied the PCC of expanded balls in Euclidean space, that is, $\ell_2^d$. We also prove lower bounds on the VC dimension that match the upper bounds in terms of the margin parameter $δ$. Finally, we derive a Dense Neighborhood Lemma in $L_p$-spaces, again extending the known Euclidean results. Our method relies on the linearization of the distance through a map into a space of non-trivial Rademacher type, and then the use of a balanced signed-sum estimate, or a no-dimensional Radon theorem. The arguments rely on ideas from functional analysis that are clearly explained for the non-expert in that field.

cs.LG↗

arXiv2025

John Ellipsoids of Revolution

Finding a largest Euclidean ball in a given convex body $K \subset \mathbb{R}^d$ and finding a largest volume ellipsoid in $K$ are two problems of fundamentally different nature. The first is a purely Euclidean problem, where we consider scaled copies of the origin-centered closed unit ball, whereas in the second problem, we search among all affine copies of the unit ball. In this paper, we interpolate between these two classical problems by considering ellipsoids of revolution. More generally, we study pairs of convex bodies $K$ and $L$, and seek a largest-volume affine image of $K$ contained within $L$, subject to certain restrictions on the allowed affine transformations. We derive first-order necessary conditions for optimality, generalizing known conditions from the unrestricted affine setting. Using these conditions, we show that an extremal ellipsoid of revolution exhibits properties analogous to those of either the largest-volume ellipsoid or the largest Euclidean ball, depending on whether the ellipsoid is considered along its axis of revolution or along the orthogonal complement of that axis.

math.MG↗

arXiv2025

Selected topics from the theory of intersections of balls

In this survey, we discuss volumetric and combinatorial results concerning (mostly finite) intersections or unions of balls (mostly of equal radii) in the $d$-dimensional real vector space, mostly equipped with the Euclidean norm. Our first topic is the Kneser--Poulsen Conjecture, according to which if a finite number of balls are rearranged so that the pairwise distances of the centers increase, then the volume of the union (resp., intersection) increases (resp., decreases). Next, we discuss Blaschke--Santaló-type inequalities, and reverse isoperimetric inequalities for convex sets in Euclidean $d$-space obtained as intersections of (possibly infinitely many) balls of radius $r$, which we call $r$-ball bodies. We present some results on $1$-ball bodies (also called ball-bodies or spindle convex sets) in the plane, with special attention paid to their approximation by the spindle convex hull of a finite subset. A ball-polyhedron is a ball-body obtained as the intersection of finitely many unit balls in Euclidean $d$-space. We consider the combinatorial structure of their faces, and volumetric properties of ball-polyhedra obtained from choosing the centers of the balls randomly.

math.MG↗

arXiv2025

Higher rank antipodality

Motivated by general probability theory, we say that the set $S$ in $\mathbb{R}^d$ is \emph{antipodal of rank $k$}, if for any $k+1$ elements $q_1,\ldots q_{k+1}\in S$, there is an affine map from $\mathrm{conv}(S)$ to the $k$-dimensional simplex $Δ_k$ that maps $q_1,\ldots q_{k+1}$ bijectively onto the $k+1$ vertices of $Δ_k$. For $k=1$, it coincides with the well-studied notion of (pairwise) antipodality introduced by Klee. We consider the following natural generalization of Klee's problem on antipodal sets: What is the maximum size of an antipodal set of rank $k$ in $\mathbb{R}^d$? We present a geometric characterization of antipodal sets of rank $k$ and adapting the argument of Danzer and Grünbaum originally developed for the $k=1$ case, we prove an upper bound which is exponential in the dimension. We show that this problem can be connected to a classical question in computer science on finding perfect hashes, and it provides a lower bound on the maximum size, which is also exponential in the dimension. By connecting rank-$k$ antipodality to $k$-neighborly polytopes, we obtain another upper bound when $k>d/2$.

math.MG↗

arXiv2023

Quantitative Steinitz theorem: A spherical version

Steinitz's theorem states that if the origin belongs to the interior of the convex hull of a set $Q \subset \mathbb{R}^d$, then there are at most $2d$ points $Q^\prime$ of $Q$ whose convex hull contains the origin in the interior. Bárány, Katchalski and Pach gave a quantitative version whereby the radius of the ball contained in the convex hull of $Q^\prime$ is bounded from below. In the present note, we show that a Euclidean result of this kind implies a corresponding spherical version.

math.MG↗

arXiv2023

Functional John and Löwner conditions for pairs of log-concave functions

John's fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ in $\mathbb{R}^d$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids by positions (affine images) of another body $L$. Another, more recent direction is to consider logarithmically concave functions on $\mathbb{R}^d$ instead of convex bodies: we designate some special, radially symmetric log-concave function $g$ as the analogue of the Euclidean ball, and want to find its largest integral position under the constraint that it is pointwise below some given log-concave function $f$. We follow both directions simultaneously: we consider the functional question, and allow essentially any meaningful function to play the role of $g$ above. Our general theorems jointly extend known results in both directions. The dual problem in the setting of convex bodies asks for the smallest volume ellipsoid, called \emph{L{ö}wner's ellipsoid}, containing $K$. We consider the analogous problem for functions: we characterize the solutions of the optimization problem of finding a smallest integral position of some log-concave function $g$ under the constraint that it is pointwise above $f$. It turns out that in the functional setting, the relationship between the John and the L{ö}wner problems is more intricate than it is in the setting of convex bodies.

math.MG↗

arXiv2023

Quantitative Steinitz Theorem: A polynomial bound

The classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are at most $2d$ points of $S$ whose convex hull contains the origin in the interior. Bárány, Katchalski, and Pach proved the following quantitative version of Steinitz's theorem. Let $Q$ be a convex polytope in $\mathbb{R}^d$ containing the standard Euclidean unit ball $\mathbf{B}^d$. Then there exist at most $2d$ vertices of $Q$ whose convex hull $Q^\prime$ satisfies \[ r \mathbf{B}^d \subset Q^\prime \] with $r\geq d^{-2d}$. They conjectured that $r\geq c d^{-1/2}$ holds with a universal constant $c>0$. We prove $r \geq \frac{1}{5d^2}$, the first polynomial lower bound on $r$. Furthermore, we show that $r$ is not be greater than $\frac{2}{\sqrt{d}}$.

math.MG↗

arXiv2023

On Helly numbers of exponential lattices

Given a set $S \subseteq \mathbb{R}^2$, define the \emph{Helly number of $S$}, denoted by $H(S)$, as the smallest positive integer $N$, if it exists, for which the following statement is true: for any finite family $\mathcal{F}$ of convex sets in~$\mathbb{R}^2$ such that the intersection of any $N$ or fewer members of~$\mathcal{F}$ contains at least one point of $S$, there is a point of $S$ common to all members of $\mathcal{F}$. We prove that the Helly numbers of \emph{exponential lattices} $\{α^n \colon n \in \mathbb{N}_0\}^2$ are finite for every $α>1$ and we determine their exact values in some instances. In particular, we obtain $H(\{2^n \colon n \in \mathbb{N}_0\}^2)=5$, solving a problem posed by Dillon (2021). For real numbers $α, β> 1$, we also fully characterize exponential lattices $L(α,β) = \{α^n \colon n \in \mathbb{N}_0\} \times \{β^n \colon n \in \mathbb{N}_0\}$ with finite Helly numbers by showing that $H(L(α,β))$ is finite if and only if $\log_α(β)$ is rational.

math.CO↗

arXiv2022

Contacts in totally separable packings in the plane and in high dimensions

We study the contact structure of totally separable} packings of translates of a convex body $K$ in $\mathbb{R}^d$, that is, packings where any two touching bodies have a separating hyperplane that does not intersect the interior of any translate in the packing. The separable Hadwiger number $H_{\text{sep}}(K)$ of $K$ is defined to be the maximum number of translates touched by a single translate, with the maximum taken over all totally separable packings of translates of $K$. We show that for each $d\geq 8$, there exists a smooth and strictly convex $K$ in $\mathbb{R}^d$ with $H_{\text{sep}}(K)>2d$, and asymptotically, $H_{\text{sep}}(K)=Ω\bigl((3/\sqrt{8})^d\bigr)$. We show that Alon's packing of Euclidean unit balls such that each translate touches at least $2^{\sqrt{d}}$ others whenever $d$ is a power of $4$, can be adapted to give a totally separable packing of translates of the $\ell_1$-unit ball with the same touching property. We also consider the maximum number of touching pairs in a totally separable packing of $n$ translates of any planar convex body $K$. We prove that the maximum equals $\lfloor 2n-2\sqrt{n}\rfloor$ if and only if $K$ is a quasi hexagon, thus completing the determination of this value for all planar convex bodies.

math.MG↗

arXiv2021

Covering convex bodies and the Closest Vector Problem

We present algorithms for the $(1+ε)$-approximate version of the closest vector problem for certain norms. The currently fastest algorithm (Dadush and Kun 2016) for general norms has running time of $2^{O(n)} (1/ε)^n$. We improve this substantially in the following two cases. For $\ell_p$-norms with $p>2$ (resp. $p \in [1,2]$) fixed, we present an algorithm with a running time of $2^{O(n)} (1/ε)^{n/2}$ (resp. $2^{O(n)} (1/ε)^{n/p}$). This result is based on a geometric covering problem, that was introduced in the context of CVP by Eisenbrand et al.: How many convex bodies are needed to cover the ball of the norm such that, if scaled by two around their centroids, each one is contained in the $(1+ε)$-scaled homothet of the norm ball? We provide upper bounds for this problem by exploiting the \emph{modulus of smoothness} of the $\ell_p$-balls. Applying a covering scheme, we can boost any constant approximation algorithm for CVP to a $(1+ε)$-approximation algorithm with the improved run time, either using a straightforward sampling routine or using the deterministic algorithm of Dadush for the construction of an epsilon net. The space requirement only depends on the constant approximation CVP solver used. Furthermore, we generalise the result of Eisenbrand et al. for the $\ell_\infty$-norm. For centrally symmetric polytopes (resp. zonotopes) with $O(n)$ facets (resp. generated by $O(n)$ line segments), we provide a deterministic $O(\log_2(1/ε))^{O(n)}$ time algorithm. Finally, we establish a connection between the \emph{modulus of smoothness} and \emph{lattice sparsification}. Using the enumeration and sparsification tools developped by Dadush, Kun, Peikert and Vempala, this leads to a simple alternative to the boosting procedure for CVP under $\ell_p$-norms. This connection might be of independent interest.

cs.DS↗

arXiv2021

Functional John Ellipsoids

We introduce a new way of representing logarithmically concave functions on $\mathbb{R}^{d}$. It allows us to extend the notion of the largest volume ellipsoid contained in a convex body to the setting of logarithmically concave functions as follows. For every $s>0$, we define a class of non-negative functions on $\mathbb{R}^{d}$ derived from ellipsoids in $\mathbb{R}^{d+1}$. For any log-concave function $f$ on $\mathbb{R}^{d}$, and any fixed $s>0$, we consider functions belonging to this class, and find the one with the largest integral under the condition that it is pointwise less than or equal to $f$, and we call it the \emph{\jsfunction} of $f$. After establishing existence and uniqueness, we give a characterization of this function similar to the one given by John in his fundamental theorem. We find that John $s$-functions converge to characteristic functions of ellipsoids as $s$ tends to zero and to Gaussian densities as $s$ tends to infinity. As an application, we prove a quantitative Helly type result: the integral of the pointwise minimum of any family of log-concave functions is at least a constant $c_d$ multiple of the integral of the pointwise minimum of a properly chosen subfamily of size $3d+2$, where $c_d$ depends only on $d$.

math.FA↗

arXiv2021

A Quantitative Helly-type Theorem: Containment in a Homothet

We introduce a new variant of quantitative Helly-type theorems: the minimal \emph{"homothetic distance"} of the intersection of a family of convex sets to the intersection of a subfamily of a fixed size. As an application, we establish the following quantitative Helly-type result for the \emph{diameter}. If $K$ is the intersection of finitely many convex bodies in $\mathbb{R}^d$, then one can select $2d$ of these bodies whose intersection is of diameter at most $(2d)^3\mathrm{diam}(K)$. The best previously known estimate, due to Brazitikos, is $c d^{11/2}$. Moreover, we confirm that the multiplicative factor $c d^{1/2}$ conjectured by Bárány, Katchalski and Pach cannot be improved.

math.MG↗

arXiv2021

Computing the covering radius of a polytope with an application to lonely runners

We are concerned with the computational problem of determining the covering radius of a rational polytope. This parameter is defined as the minimal dilation factor that is needed for the lattice translates of the correspondingly dilated polytope to cover the whole space. As our main result, we describe a new algorithm for this problem, which is simpler, more efficient and easier to implement than the only prior algorithm of Kannan (1992). Motivated by a variant of the famous Lonely Runner Conjecture, we use its geometric interpretation in terms of covering radii of zonotopes, and apply our algorithm to prove the first open case of three runners with individual starting points.

math.CO↗

arXiv2021

Quantitative Fractional Helly and $(p,q)$-Theorems

We consider quantitative versions of Helly-type questions, that is, instead of finding a point in the intersection, we bound the volume of the intersection. Our first main geometric result is a quantitative version of the Fractional Helly Theorem of Katchalski and Liu, the second one is a quantitative version of the $(p,q)$-Theorem of Alon and Kleitman.

math.MG↗

arXiv2020

An analogue of a theorem of Steinitz for ball polyhedra in $\mathbb{R}^3$

Steinitz's theorem states that a graph $G$ is the edge-graph of a $3$-dimensional convex polyhedron if and only if, $G$ is simple, plane and $3$-connected. We prove an analogue of this theorem for ball polyhedra, that is, for intersections of finitely many unit balls in $\mathbb{R}^3$.

math.MG↗

arXiv2020

Colorful Helly-type Theorems for the Volume of Intersections of Convex Bodies

We prove the following Helly-type result. Let $\mathcal{C}_1,\dots,\mathcal{C}_{3d}$ be finite families of convex bodies in $\mathbb{R}^d$. Assume that for any colorful selection of $2d$ sets, $C_{i_k}\in \mathcal{C}_{i_k}$ for each $1\leq k\leq 2d$ with $1\leq i_1<\dots<i_{2d}\leq 3d$, the intersection $\bigcap\limits_{k=1}^{2d} C_{i_k}$ is of volume at least 1. Then there is an $1\leq i \leq 3d$ such that $\bigcap\limits_{C\in \mathcal{C}_i} C$ is of volume at least $d^{-O(d^2)}$.

math.MG↗

arXiv2020

Approximation of the average of some random matrices

Rudelson's theorem states that if for a set of unit vectors $u_i$ and positive weights $c_i$, we have that $\sum c_i u_i\otimes u_i$ is the identity operator $I$ on ${\mathbb R}^d$, then the sum of a random sample of $Cd\ln d$ of these diadic products is close to $I$. The $\ln d$ term cannot be removed. On the other hand, the recent fundamental result of Batson, Spielman and Srivastava and its improvement by Marcus, Spielman and Srivastava show that the $\ln d$ term can be removed, if one wants to show the existence of a good approximation of $I$ as the average of a few diadic products. It is known that essentially the same proof as Rudelson's yields a more general statement about the average of positive semi-definite matrices. First, we give an example of an average of positive semi-definite matrices where there is no approximation of this average by $Cd$ elements. Thus, the result of Batson, Spielman and Srivastava cannot be extended to this wider class of matrices. Next, we present a stability version of Rudelson's result on positive semi-definite matrices, and thus, extend it to certain non-symmetric matrices. This yields applications to the study of the Banach--Mazur distance of convex bodies. Finally, we show that in some cases, one needs to take a subset of the vectors of order $d^2$ to approximate the identity.

math.FA↗

arXiv2019

Angular measures and Birkhoff orthogonality in Minkowski planes

Let $x$ and $y$ be two unit vectors in a normed plane $\mathbb{R}^2$. We say that $x$ is Birkhoff orthogonal to $y$ if the line through $x$ in the direction $y$ supports the unit disc. A B-measure (Fankhänel 2011) is an angular measure $μ$ on the unit circle for which $μ(C)=π/2$ whenever $C$ is a shorter arc of the unit circle connecting two Birkhoff orthogonal points. We present a characterization of the normed planes that admit a B-measure.

math.MG↗