arXiv subjects

Zsolt Lángi

Publications and source records attributed to Zsolt Lángi.

At least 19 recordsLinked to original sources

arXiv2026

Covering a square by congruent squares

The main goal of this paper is to address the following problem: given a positive integer $n$, find the largest value $S(n)$ such that a square of edge length $S(n)$ in the Euclidean plane can be covered by $n$ unit squares. We investigate also the variant in which the goal is to cover only the boundary of a square. We show that these two problems are equivalent for $n \leq 4$, but not for $n=5$. For both problems, we also present the solutions for $n=5$.

math.MG↗

arXiv2025

Notes on non-separable arrangements of convex bodies

A problem posed by Erdős in 1945 initiated the study of non-separable arrangements of convex bodies. A finite collection of convex bodies in Euclidean $d$-space is called a non-separable family (or NS-family) if every hyperplane intersecting their convex hull also intersects at least one member of the family. Recent work has focused on minimal coverings of NS-families consisting of positive homothetic convex bodies. In this paper, we strengthen these results by establishing their analogues for weakly non-separable families of convex polytopes. We further obtain stability results and analyze maximal weakly non-separable families of cubes. As an additional extension, we also examine weakly $k$-impassable families of convex $d$-polytopes for $0<k<d-1$.

math.MG↗

arXiv2025

The Honeycomb Conjecture in normed planes and an alpha-convex variant of a theorem of Dowker

The Honeycomb Conjecture states that among tilings with unit area cells in the Euclidean plane, the average perimeter of a cell is minimal for a regular hexagonal tiling. This conjecture was proved by L. Fejes Tóth for convex tilings, and by Hales for not necessarily convex tilings. In this paper we investigate the same question for tilings of a given normed plane, and show that among normal, convex tilings in a normed plane, the average squared perimeter of a cell is minimal for a tiling whose cells are translates of a centrally symmetric hexagon. We also show that the question whether the same statement is true for the average perimeter of a cell is closely related to an $α$-convex variant of a theorem of Dowker on the area of polygons circumscribed about a convex disk. Exploring this connection we find families of norms in which the average perimeter of a cell of a tiling is minimal for a hexagonal tiling, and prove some additonal related results. Finally, we apply our method to give a partial answer to a problem of Steinhaus about the isoperimetric ratios of cells of certain tilings in the Euclidean plane, appeared in an open problem book of Croft, Falconer and Guy.

math.MG↗

arXiv2025

On the edge densities of normal, convex mosaics

In this paper we investigate the problem of finding the minimum edge density in families of convex, normal mosaics with unit volume cells in $n$-dimensional Euclidean space. In the first part of the paper we solve this problem for mosaics whose cells are Minkowski sums of cells of $1$ or $2$-dimensional mosaics. We show that while for $n=2$ this minimum is attained by a mosaic with regular hexagon cells, this is not true in any dimension $n > 2$, where the minimum is attained by a mosaic whose cells are Minkowski sums of pairwise orthogonal regular triangles, and possibly a segment. In the second part we investigate $3$-dimensional convex mosaics whose cells are translates of a given convex polyhedron, and show that within this family, mosaics with cubes as cells have minimum edge density. In addition, using our method, in the family of $3$-dimensional convex polyhedra whose translates tile the space, we find the unit volume polyhedra with minimal total edge length.

math.MG↗

arXiv2025

Curves with increasing chords in normed planes

A curve has the increasing chord property if for any points $a,b,c,d$ in this order on the curve, the distance of $a,d$ is not smaller than that of $b,c$. Answering a conjecture of Larman and McMullen, Rote proved in 1994 that the arclength of a curve in the Euclidean plane with the increasing chord property is at most $\frac{2π}{3}$ times the distance of its endpoints, and this inequality is sharp. In this note we generalize the result of Rote for curves in a normed plane with a strictly convex norm, based on an investigation of the geometric properties of involutes in normed planes. We also discuss some related extremum problems.

math.MG↗

arXiv2025

Arcs with increasing chords in $\mathbf{R}^d$

A curve $γ$ that connects $s$ and $t$ has the increasing chord property if $|bc| \leq |ad|$ whenever $a,b,c,d$ lie in that order on $γ$. For planar curves, the length of such a curve is known to be at most $2π/3 \cdot |st|$. Here we examine the question in higher dimensions and from the algorithmic standpoint and show the following: (I) The length of any $s-t$ curve with increasing chords in $\mathbf{R}^d$ is at most $2 \cdot \left( e/2 \cdot (d+4) \right)^{d-1} \cdot |st|$ for every $d \geq 3$. This is the first bound in higher dimensions. (II) Given a polygonal chain $P=(p_1, p_2, \dots, p_n)$ in $\mathbf{R}^d$, where $d \geq 4$, $k =\lfloor d/2 \rfloor$, it can be tested whether it satisfies the increasing chord property in $O\left(n^{2-1/(k+1)} {\rm polylog} (n) \right)$ expected time. This is the first subquadratic algorithm in higher dimensions.

cs.CG↗

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

Steiner symmetrization on the sphere

The aim of this paper is to introduce a generalization of Steiner symmetrization in Euclidean space for spherical space, which is the dual of the Steiner symmetrization in hyperbolic space introduced by J. Schneider (Manuscripta Math. 60: 437-461, 1988). We show that this symmetrization preserves volume in every dimension, and convexity in the spherical plane, but not in dimensions $n > 2$. In addition, we investigate the monotonicity properties of the perimeter and diameter of a set under this process, and find conditions under which the image of a spherically convex disk under a suitable sequence of Steiner symmetrizations converges to a spherical cap. We apply our results to prove a spherical analogue of a theorem of Sas, and to confirm a conjecture of Besau and Werner (Adv. Math. 301: 867-901, 2016) for centrally symmetric spherically convex disks. Lastly, we prove a spherical variant of a theorem of Winternitz.

math.MG↗

arXiv2025

Centroids and equilibrium points of convex bodies

The aim of this note is to survey the results in some geometric problems related to the centroids and the static equilibrium points of convex bodies. In particular, we collect results related to Grünbaum's inequality and the Busemann-Petty centroid inequality, describe classifications of convex bodies based on equilibrium points, and investigate the location and structure of equilibrium points, their number with respect to a general reference point as well as the static equilibrium properties of convex polyhedra.

math.MG↗

arXiv2025

On separability in discrete geometry

A problem of Erdős (Amer. Math. Monthly 52: 494-498, 1945) and a theorem of Fejes Tóth and Fejes Tóth (Acta Math. Acad. Sci. Hungar. 24: 229-232, 1973) initiated the study of non-separable arrangements of convex bodies and the investigation of totally separable packings of convex bodies with both topics analyzing the concept of separability from the point view of discrete geometry. This article surveys the progress made on these and some closely related problems and highlights the relevant questions that have been left open.

math.MG↗

arXiv2024

Density bounds for unit ball packings relative to their outer parallel domains

We prove that the highest density of non-overlapping translates of a given centrally symmetric convex domain relative to its outer parallel domain of given outer radius is attained by a lattice packing in the Euclidean plane. This generalizes some earlier (classical) results. Sharp upper bounds are proved for the analogue problem on congruent circular disks in the spherical (resp., hyperbolic) plane and on congruent balls in Euclidean $3$-space.

math.MG↗

arXiv2024

Remarks on soft ball packings in dimensions 2 and 3

We study translative arrangements of centrally symmetric convex domains in the plane (resp., of congruent balls in the Euclidean $3$-space) that neither pack nor cover. We define their soft density depending on a soft parameter and prove that the largest soft density for soft translative packings of a centrally symmetric convex domain with $3$-fold rotational symmetry and given soft parameter is obtained for a proper soft lattice packing. Furthermore, we show that among the soft lattice packings of congruent soft balls with given soft parameter the soft density is locally maximal for the corresponding face centered cubic (FCC) lattice.

math.MG↗

arXiv2024

Dowker-type theorems for disk-polygons in normed planes

A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body $K$ in the Euclidean plane, the areas of the maximum (resp. minimum) area convex $n$-gons inscribed (resp. circumscribed) in $K$ is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, the Euclidean plane by an arbitrary normed plane, or convex $n$-gons by disk-$n$-gons, obtained as the intersection of $n$ closed Euclidean unit disks. The aim of our paper is to investigate these problems for $C$-$n$-gons, defined as intersections of $n$ translates of the unit disk $C$ of a normed plane. In particular, we show that Dowker's theorem remains true for the areas and the perimeters of circumscribed $C$-$n$-gons, and the perimeters of inscribed $C$-$n$-gons. We also show that in the family of origin-symmetric plane convex bodies, for a typical element $C$ with respect to Hausdorff distance, Dowker's theorem for the areas of inscribed $C$-$n$-gons fails.

math.MG↗

arXiv2024

An algorithm to find maximum area polygons circumscribed about a convex polygon

A convex polygon Q is circumscribed about a convex polygon P if every vertex of P lies on at least one side of Q. We present an algorithm for finding a maximum area convex polygon circumscribed about any given convex n-gon in O(n^3) time. As an application, we disprove a conjecture of Farris. Moreover, for the special case of regular n-gons we find an explicit solution.

math.MG↗

arXiv2024

On a Dowker-type problem for convex disks with almost constant curvature

A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body $K$, the areas of the maximum (resp. minimum) area convex $n$-gons inscribed (resp. circumscribed) in $K$ is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, or convex $n$-gons by disk-$n$-gons, obtained as the intersection of $n$ closed Euclidean unit disks. It has been proved recently that if $C$ is the unit disk of a normed plane, then the same properties hold for the area of $C$-$n$-gons circumscribed about a $C$-convex disk $K$ and for the perimeters of $C$-$n$-gons inscribed or circumscribed about a $C$-convex disk $K$, but for a typical origin-symmetric convex disk $C$ with respect to Hausdorff distance, there is a $C$-convex disk $K$ such that the sequence of the areas of the maximum area $C$-$n$-gons inscribed in $K$ is not concave. The aim of this paper is to investigate this question if we replace the topology induced by Hausdorff distance with a topology induced by the surface area measure of the boundary of $C$.

math.MG↗

arXiv2023

On monohedral tilings of a regular polygon

A tiling of a topological disc by topological discs is called monohedral if all tiles are congruent. Maltby (J. Combin. Theory Ser. A 66: 40-52, 1994) characterized the monohedral tilings of a square by three topological discs. Kurusa, Lángi and V\'ıgh (Mediterr. J. Math. 17: article number 156, 2020) characterized the monohedral tilings of a circular disc by three topological discs. The aim of this note is to connect these two results by characterizing the monohedral tilings of any regular $n$-gon with at most three tiles for any $n \geq 5$.

math.MG↗

arXiv2023

Morse-Smale complexes on convex polyhedra

Motivated by applications in geomorphology, the aim of this paper is to extend Morse-Smale theory from smooth functions to the radial distance function (measured from an internal point), defining a convex polyhedron in 3-dimensional Euclidean space. The resulting polyhedral Morse-Smale complex may be regarded, on one hand, as a generalization of the Morse-Smale complex of the smooth radial distance function defining a smooth, convex body, on the other hand, it could be also regarded as a generalization of the Morse-Smale complex of the piecewise linear parallel distance function (measured from a plane), defining a polyhedral surface. Beyond similarities, our paper also highlights the marked differences between these three problems and it also relates our theory to other methods. Our work includes the design, implementation and testing of an explicit algorithm computing the Morse-Smale complex on a convex polyhedron.

cs.CG↗