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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 unit 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-Santal\'o type, and isoperimetric inequalities for convex sets in Euclidean $d$-space obtained as intersections of (possibly infinitely many) unit balls, which we call spindle convex sets. We present some results on 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 convex 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., Comment: 30 pages

A problem of Erd\H{o}s (Amer. Math. Monthly 52: 494-498, 1945) and a theorem of Fejes T\'oth and Fejes T\'oth (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., Comment: 28 pages, 10 figures

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 Gr\"unbaum'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., Comment: 27 pages, 1 table, 5 figures

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 T\'oth 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 $\alpha$-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., Comment: 20 pages, 3 figures

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., Comment: 28 pages, 3 figures

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., Comment: 13 pages, 4 figures

In this paper we investigate the problem of finding the minimum of edge density in families of convex mosaics with unit volume cells in $3$-dimensional Euclidean space. We solve this problem for two families. In particular, we prove that a translative convex mosaic with unit volume cells has minimal edge density if it is a face-to-face mosaic generated by unit cubes. We also investigate decomposable mosaics, defined as mosaics whose cells are Minkowski sums of cells of two lower dimensional convex mosaics, and show that the edge density of every normal decomposable mosaic with unit volume cells is at least as large as that of a face-to-face mosaic with regular triangle-based right prisms as cells., Comment: 17 pages, 1 figure, 1 table. arXiv admin note: text overlap with arXiv:2102.11621

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., Comment: 11 pages, 3 figures

The purpose of this technical appendix is to share more detailed results and to describe more fully the sample and methods used in the research included in the brief, "The Widening Achievement Divide during COVID-19." The authors investigated two main research questions in this brief: (1) To what degree have students' reading and math test scores become more variable as a result of the COVID-19 pandemic?; and (2) How do achievement gains across the pandemic compare to pre-pandemic trends for students who were low- or high-achieving at the start of the pandemic? [For the full report, see ED627139.]

In this brief we examine how variability in students' math and reading test scores changed across the pandemic, including whether increased dispersion was concentrated at one part of the test score distribution. Using reading and math assessment data from a sample of 8 million students in grades 3-8 in 24,000 public schools who took MAP Growth assessments in spring 2022 and a similar sized sample from before the pandemic, we examined the degree to which test scores became more variable. To understand how impacts may have differed for students with the highest and lowest test scores before the pandemic, we followed a subset of 1 million students whose pre-pandemic achievement placed them in the top or bottom decile of the achievement distribution relative to national pre-pandemic norms. Initial findings show students have a wider range of achievement levels in spring 2022 than would be expected in a typical school year, and that this increased variability in test scores is primarily driven by a spreading out at the bottom end of the distribution, with test scores for lower-achieving students dropping more than the declines observed for higher-achieving students. Low achievers experienced larger initial impacts at the onset of the pandemic and less improvement during 2021-22. These results highlight a key challenge for teachers who must now meet the needs of students who may be in the same grade but differ even more than they did prior to the pandemic in terms of learning needs, and suggest a critical need for focusing support to help the lowest achieving students catch up to address the achievement divide that has widened significantly over the last two years. [For the technical brief, see ED627140.]

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 $C$., Comment: 13 pages, 3 figures

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., Comment: 21 pages, 5 figures

Let $\mathcal{P}$ be a packing of circular disks of radius $\rho>0$ in the Euclidean, spherical, or hyperbolic plane. Let $0\leq\lambda\leq\rho$. We say that $\mathcal{P}$ is a $\lambda$-separable packing of circular disks of radius $\rho$ if the family $\mathcal{P'}$ of disks concentric to the disks of $\mathcal{P}$ having radius $\lambda$ form a totally separable packing, i.e., any two disks of $\mathcal{P'}$ can be separated by a line which is disjoint from the interior of every disk of $\mathcal{F'}$. This notion bridges packings of circular disks of radius $\rho$ (with $\lambda=0$) and totally separable packings of circular disks of radius $\rho$ (with $\lambda=\rho$). In this note we extend several theorems on the density, tightness, and contact numbers of disk packings and totally separable disk packings to $\lambda$-separable packings of circular disks of radius $\rho$ in the Euclidean, spherical, and hyperbolic plane. In particular, our upper bounds (resp., lower bounds) for the density (resp., tightness) of $\lambda$-separable packings of unit disks in the Euclidean plane are sharp for all $0\leq\lambda\leq 1$ with the extremal values achieved by $\lambda$-separable lattice packings of unit disks. On the other hand, the bounds of similar results in the spherical and hyperbolic planes are not sharp for all $0\leq\lambda\leq\rho$ although they do not seem to be far from the relevant optimal bounds either. The proofs use local analytic and elementary geometry and are based on the so-called refined Moln\'ar decomposition, which is obtained from the underlying Delaunay decomposition and as such might be of independent interest., Comment: 20 pages, 6 figures

The aim of this paper is to prove isoperimetric inequalities for simplices and polytopes with $d+2$ vertices in Euclidean, spherical and hyperbolic $d$-space. In particular, we find the minimal volume $d$-dimensional hyperbolic simplices and spherical tetrahedra of a given inradius. Furthermore, we investigate the properties of maximal volume spherical and hyperbolic polytopes with $d+2$ vertices with a given circumradius, and the hyperbolic polytopes with $d+2$ vertices with a given inradius and having a minimal volume or minimal total edge length. Finally, for any $1 \leq k \leq d$, we investigate the properties of Euclidean simplices and polytopes with $d+2$ vertices having a fixed inradius and a minimal volume of its $k$-skeleton. The main tool of our investigation is Euclidean, spherical and hyperbolic Steiner symmetrization., Comment: 17 pages, 3 figures

Shephard (Canad. J. Math. 26: 302-321, 1974) proved a decomposition theorem for zonotopes yielding a simple formula for their volume. In this note we prove a generalization of this theorem yielding similar formulas for their intrinsic volumes. We use this result to investigate geometric extremum problems for zonotopes generated by a given number of segments. In particular, we solve isoperimetric problems for d-dimensional zonotopes generated by d or d+1 segments, and give asymptotic estimates for the solutions of similar problems for zonotopes generated by sufficiently many segments. In addition, we present applications of our results to the \ell_1$ polarization problem on the unit sphere and to a vector-valued Maclaurin inequality conjectured by Brazitikos and McIntyre in 2021., Comment: 24 pages, 2 figures

A family of spherical caps of the 2-dimensional unit sphere $\mathbb{S}^2$ is called a totally separable packing in short, a TS-packing if any two spherical caps can be separated by a great circle which is disjoint from the interior of each spherical cap in the packing. The separable Tammes problem asks for the largest density of given number of congruent spherical caps forming a TS-packing in $\mathbb{S}^2$. We solve this problem up to $8$ spherical caps and upper bound the density of any TS-packing of congruent spherical caps in terms of their angular radius. Based on this, we show that the centered separable kissing number of $3$-dimensional Euclidean balls is $8$. Furthermore, we prove bounds for the maximum of the smallest inradius of the cells of the tilings generated by $n>1$ great circles in $\mathbb{S}^2$. Next, we prove dual bounds for TS-coverings of $\mathbb{S}^2$ by congruent spherical caps. Here a covering of $\mathbb{S}^2$ by spherical caps is called a totally separable covering in short, a TS-covering if there exists a tiling generated by finitely many great circles of $\mathbb{S}^2$ such that the cells of the tiling are covered by pairwise distinct spherical caps of the covering. Finally, we extend some of our bounds on TS-coverings to spherical spaces of dimension $>2$., Comment: Theorem 3 and its proof on pages 17-19, the footnote on page 7, and the quotation of Glazyrin's theorem on page 26 are new. Overall: 28 pages and 22 figures

Answering a question of Conway and Guy in a 1968 paper, L\'angi in 2021 proved the existence of a monostable polyhedron with $n$-fold rotational symmetry for any $n \geq 3$, and arbitrarily close to a Euclidean ball. In this paper we strengthen this result by characterizing the possible symmetry groups of all mono-monostatic smooth convex bodies and convex polyhedra. Our result also answers a stronger version of the question of Conway and Guy, asked in the above paper of L\'angi., Comment: 17 pages, 6 figures

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, L\'angi and V\'\i gh (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 $n \geq 5$., Comment: 18 pages, 9 figures

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., Comment: 25 pages, 9 figures

The aim of this note is to investigate isoperimetric-type problems for $3$-dimensional parallelohedra; that is, for convex polyhedra whose translates tile the $3$-dimensional Euclidean space. Our main result states that among $3$-dimensional parallelohedra with unit volume the one with minimal mean width is the regular truncated octahedron., Comment: Corrected a minor error in Remark 2. 11 pages, 1 figure, 1 table

The concept of a Minkowski arrangement was introduced by Fejes T\'oth in 1965 as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by Fejes T\'oth in 1967, who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order $\mu$ for some $0 < \mu < 1$ if no member $K$ of the family overlaps the homothetic copy of any other member $K'$ with ratio $\mu$ and with the same center as $K'$. In this note we prove a sharp upper bound on the total area of the elements of a generalized Minkowski arrangement of order $\mu$ of finitely many circular disks in the Euclidean plane. This result is a common generalization of a similar result of Fejes T\'oth for Minkowski arrangements of circular disks, and a result of B\"or\"oczky and Szab\'o about the maximum density of a generalized Minkowski arrangement of circular disks in the plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski arrangement of homothetic copies of a centrally symmetric convex body., Comment: 17 pages, 6 figures

A convex polygon $Q$ is inscribed in a convex polygon $P$ if every side of $P$ contains at least one vertex of $Q$. We present algorithms for finding a minimum area and a minimum perimeter convex polygon inscribed in any given convex $n$-gon in $O(n)$ and $O(n^3)$ time, respectively. We also investigate other variants of this problem., Comment: 15 pages, 4 figures

The aim of this note is to investigate the properties of the convex hull and the homothetic convex hull functions of a convex body $K$ in Euclidean $n$-space, defined as the volume of the union of $K$ and one of its translates, and the volume of $K$ and a translate of a homothetic copy of $K$, respectively, as functions of the translation vector. In particular, we prove that the convex hull function of the body $K$ does not determine $K$. Furthermore, we prove the equivalence of the polar projection body problem raised by Petty, and a conjecture of G.Horv\'ath and L\'angi about translative constant volume property of convex bodies. We give a short proof of some theorems of Jer\'onimo-Castro about the homothetic convex hull function, and prove a homothetic variant of the translative constant volume property conjecture for $3$-dimensional convex polyhedra. We also apply our results to describe the properties of the illumination bodies of convex bodies., Comment: 12 pages, 1 figure

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 $\mathbb{R}^3$., Comment: 9 pages

A convex polyhedron is called monostable if it can rest in stable position only on one of its faces. The aim of this paper is to investigate three questions of Conway, regarding monostable polyhedra, which first appeared in a 1969 paper of Goldberg and Guy (M. Goldberg and R.K. Guy, Stability of polyhedra (J.H. Conway and R.K. Guy), SIAM Rev. 11 (1969), 78-82). In this note we answer two of these problems and make a conjecture about the third one. The main tool of our proof is a general theorem describing approximations of smooth convex bodies by convex polyhedra in terms of their static equilibrium points. As another application of this theorem, we prove the existence of a convex polyhedron with only one stable and one unstable point., Comment: 17 pages, 6 figures

The structure of $k$-diametral point configurations in Minkowski $d$-space is shown to be closely related to the properties of $k$-antipodal point configurations in $\mathbb{R}^d$. In particular, the maximum size of $k$-diametral point configurations of Minkowski $d$-spaces is obtained for given $k\geq 2$ and $d\geq 2$ generalizing Petty's results (Proc. Am. Math. Soc. 29: 369-374, 1971) on equilateral sets in Minkowski spaces. Furthermore, bounds are derived for the maximum size of $k$-diametral point configurations in Euclidean $d$-space. In the proofs convexity methods are combined with volumetric estimates and combinatorial properties of diameter graphs., Comment: 13 pages, 2 figures

An irregular vertex in a tiling by polygons is a vertex of one tile and belongs to the interior of an edge of another tile. In this paper we show that for any integer $k\geq 3$, there exists a normal tiling of the Euclidean plane by convex hexagons of unit area with exactly $k$ irregular vertices. Using the same approach we show that there are normal edge-to-edge tilings of the plane by hexagons of unit area and exactly $k$ many $n$-gons ($n>6$) of unit area. A result of Akopyan yields an upper bound for $k$ depending on the maximal diameter and minimum area of the tiles. Our result complements this with a lower bound for the extremal case, thus showing that Akopyan's bound is asymptotically tight., Comment: 7 pages, 5 figures

For a compact set $A \subset {\mathbb R}^d$ and an integer $k\ge 1$, let us denote by $$ A[k] = \left\{a_1+\cdots +a_k: a_1, \ldots, a_k\in A\right\}=\sum_{i=1}^k A$$ the Minkowski sum of $k$ copies of $A$. A theorem of Shapley, Folkmann and Starr (1969) states that $\frac{1}{k}A[k]$ converges to the convex hull of $A$ in Hausdorff distance as $k$ tends to infinity. Bobkov, Madiman and Wang (2011) conjectured that the volume of $\frac{1}{k}A[k]$ is non-decreasing in $k$, or in other words, in terms of the volume deficit between the convex hull of $A$ and $\frac{1}{k}A[k]$, this convergence is monotone. It was proved by Fradelizi, Madiman, Marsiglietti and Zvavitch (2016) that this conjecture holds true if $d=1$ but fails for any $d \geq 12$. In this paper we show that the conjecture is true for any star-shaped set $A \subset {\mathbb R}^d$ for $d=2$ and $d=3$ and also for arbitrary dimensions $d \ge 4$ under the condition $k \ge (d-1)(d-2)$. In addition, we investigate the conjecture for connected sets and present a counterexample to a generalization of the conjecture to the Minkowski sum of possibly distinct sets in ${\mathbb R}^d$, for any $d \geq 7$.

In this note we prove that any monohedral tiling of the closed circular unit disc with $k \leq 3$ topological discs as tiles has a $k$-fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain the center, and the first step towards answering a question of Stein appearing in the problem book of Croft, Falconer and Guy in 1994., Comment: submitted

In this note we introduce the problem of illumination of convex bodies in spherical spaces and solve it for a large subfamily of convex bodies. We derive from it a combinatorial version of the classical illumination problem for convex bodies in Euclidean spaces as well as a solution to that for a large subfamily of convex bodies, which in dimension three leads to special Koebe polyhedra., Comment: 7 pages

In a convex mosaic in $\mathbb{R} ^d$ we denote the average number of vertices of a cell by $\bar v$ and the average number of cells meeting at a node by $\bar n$. Except for the $d=2$ planar case, there is no known formula prohibiting points in any range of the $[\bar n, \bar v]$ plane (except for the unphysical $\bar n, \bar v < d+1$ strips). Nevertheless, in $d=3$ dimensions if we plot the 28 points corresponding to convex uniform honeycombs, the 28 points corresponding to their duals and the 3 points corresponding to Poisson-Voronoi, Poisson-Delaunay and random hyperplane mosaics, then these points appear to accumulate on a narrow strip of the $[\bar n, \bar v]$ plane. To explore this phenomenon we introduce the harmonic degree $\bar h= \bar n\bar v/(\bar n + \bar v)$ of a $d$-dimensional mosaic. We show that the observed narrow strip on the $[\bar n, \bar v]$ plane corresponds to a narrow range of $\bar h$. We prove that for every $\bar h^{\star} \in (d, 2^{d-1}]$ there exists a convex mosaic with harmonic degree $\bar h^{\star}$ and we conjecture that there exist no $d$-dimensional mosaic outside this range. We also show that the harmonic degree has deeper geometric interpretations. In particular, in case of Euclidean mosaics it is related to the average of the sum of vertex angles and their polars, and in case of 2D mosaics, it is related to the average excess angle., Comment: 20 pages, 3 figures

We define the mechanical complexity $C(P)$ of a convex polyhedron $P,$ interpreted as a homogeneous solid, as the difference between the total number of its faces, edges and vertices and the number of its static equilibria, and the mechanical complexity $C(S,U)$ of primary equilibrium classes $(S,U)^E$ with $S$ stable and $U$ unstable equilibria as the infimum of the mechanical complexity of all polyhedra in that class. We prove that the mechanical complexity of a class $(S,U)^E$ with $S, U > 1$ is the minimum of $2(f+v-S-U)$ over all polyhedral pairs $(f,v )$, where a pair of integers is called a polyhedral pair if there is a convex polyhedron with $f$ faces and $v$ vertices. In particular, we prove that the mechanical complexity of a class $(S,U)^E$ is zero if, and only if there exists a convex polyhedron with $S$ faces and $U$ vertices. We also give asymptotically sharp bounds for the mechanical complexity of the monostatic classes $(1,U)^E$ and $(S,1)^E$, and offer a complexity-dependent prize for the complexity of the G\"omb\"oc-class $(1,1)^E$., Comment: 29 pages, 13 figures

A variant of the Circle Packing Theorem states that the combinatorial class of any convex polyhedron contains elements midscribed to the unit sphere centered at the origin, and that these representatives are unique up to M\"obius transformations of the sphere. Motivated by this result, various papers investigate the problem of centering spherical configurations under M\"obius transformations. In particular, Springborn proved that for any discrete point set on the sphere there is a M\"obius transformation that maps it into a set whose barycenter is the origin, which implies that the combinatorial class of any convex polyhedron contains an element midsribed to a sphere with the additional property that the barycenter of the points of tangency is the center of the sphere. This result was strengthened by Baden, Krane and Kazhdan who showed that the same idea works for any reasonably nice measure defined on the sphere. The aim of the paper is to show that Springborn's statement remains true if we replace the barycenter of the tangency points by many other polyhedron centers. The proof is based on the investigation of the topological properties of the integral curves of certain vector fields defined in hyperbolic space. We also show that most centers of Koebe polyhedra cannot be obtained as the center of a suitable measure defined on the sphere., Comment: 21 pages, 4 figures

In recent years it became apparent that geophysical abrasion can be well characterized by the time evolution $N(t)$ of the number $N$ of static balance points of the abrading particle. Static balance points correspond to the critical points of the particle's surface represented as a scalar distance function $r$, measured from the center of mass, so their time evolution can be expressed as $N(r(t))$. The mathematical model of the particle can be constructed on two scales: on the macro scale the particle may be viewed as a smooth, convex manifold described by the smooth distance function $r$ with $N=N(r)$ equilibria, while on the micro scale the particle's natural model is a finely discretized, convex polyhedral approximation $r^{\Delta}$ of $r$, with $N^{\Delta}=N(r^{\Delta})$ equilibria. There is strong intuitive evidence suggesting that under some particular evolution models $N(t)$ and $N^{\Delta}(t)$ primarily evolve in the opposite manner. Here we create the mathematical framework necessary to understand these phenomenon more broadly, regardless of the particular evolution equation. We study micro and macro events in one-parameter families of curves and surfaces, corresponding to bifurcations triggering the jumps in $N(t)$ and $N^{\Delta}(t)$. We show that the intuitive picture developed for curvature-driven flows is not only correct, it has universal validity, as long as the evolving surface $r$ is smooth. In this case, bifurcations associated with $r$ and $r^{\Delta}$ are coupled to some extent: resonance-like phenomena in $N^{\Delta}(t)$ can be used to forecast downward jumps in $N(t)$ (but not upward jumps). Beyond proving rigorous results for the $\Delta \to 0$ limit on the nontrivial interplay between singularities in the discrete and continuum approximations we also show that our mathematical model is structurally stable, i.e. it may be verified by computer simulations., Comment: 29 pages, 7 figures

We show that under the Eikonal abrasion model, prescribing uniform normal speed in the direction of the inward surface normal, the isoperimetric quotient of a convex shape is decreasing monotonically., Comment: The proof of Theorem 1 holds only under additional assumptions. For more details, see arXiv:1910.05367

A packing of translates of a convex body in the $d$-dimensional Euclidean space $\mathbb{E}^d$ is said to be totally separable if any two packing elements can be separated by a hyperplane of $\mathbb{E}^{d}$ disjoint from the interior of every packing element. We call the packing $\mathcal P$ of translates of a centrally symmetric convex body $\mathbf{C}$ in $\mathbb{E}^d$ a $\rho$-separable packing for given $\rho\geq 1$ if in every ball concentric to a packing element of $\mathcal P$ having radius $\rho$ (measured in the norm generated by $\mathbf{C}$) the corresponding sub-packing of $\mathcal P$ is totally separable. The main result of this paper is the following theorem. Consider the convex hull $\mathbf{Q}$ of $n$ non-overlapping translates of an arbitrary centrally symmetric convex body $\mathbf{C}$ forming a $\rho$-separable packing in $\mathbb{E}^d$ with $n$ being sufficiently large for given $\rho\geq 1$. If $\mathbf{Q}$ has minimal mean $i$-dimensional projection for given $i$ with $1\leq i

A packing of translates of a convex domain in the Euclidean plane is said to be totally separable if any two packing elements can be separated by a line disjoint from the interior of every packing element. This notion was introduced by G. Fejes T\'oth and L. Fejes T\'oth (1973) and has attracted significant attention. In this paper we prove an analogue of Oler's inequality for totally separable translative packings of convex domains and then we derive from it some new results. This includes finding the largest density of totally separable translative packings of an arbitrary convex domain and finding the smallest area convex hull of totally separable packings (resp., totally separable soft packings) generated by given number of translates of a convex domain (resp., soft convex domain). Finally, we determine the largest covering ratio (that is, the largest fraction of the plane covered by the soft disks) of an arbitrary totally separable soft disk packing with given soft parameter., Comment: 19 pages, 8 figures

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., Comment: In this version we correct an error in Theorem 4 of the published version

In 1970, Coxeter gave a short and elegant geometric proof showing that if $p_1, p_2, \ldots, p_n$ are vertices of an $n$-gon $P$ in cyclic order, then $P$ is affinely regular if, and only if there is some $\lambda \geq 0$ such that $p_{j+2}-p_{j-1} = \lambda (p_{j+1}-p_j)$ for $j=1,2,\ldots, n$. The aim of this paper is to examine the properties of polygons whose vertices $p_1,p_2,\ldots,p_n \in \mathbb{C}$ satisfy the property that $p_{j+m_1}-p_{j+m_2} = w (p_{j+k}-p_j)$ for some $w \in \mathbb{C}$ and $m_1,m_2,k \in \mathbb{Z}$. In particular, we show that in `most' cases this implies that the polygon is affinely regular, but in some special cases there are polygons which satisfy this property but are not affinely regular. The proofs are based on the use of linear algebraic and number theoretic tools. In addition, we apply our method to characterize polytopes with certain symmetry groups., Comment: 11 pages, 1 figure

In this note we describe a discrete dynamical system acting on the similarity classes of a plane convex body within the affine class of the body. We find invariant elements in all affine classes, and describe the orbits of bodies in some special classes. We point out applications with abrasion processes of pebble shapes., Comment: 17 pages, 4 figures