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.Horváth and Lángi about translative constant volume property of convex bodies. We give a short proof of some theorems of Jerónimo-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.
Let $K$ be a compact convex set and $m$ be a positive integer. The covering functional of $K$ with respect to $m$ is the smallest $λ\in[0,1]$ such that $K$ can be covered by $m$ translates of $λK$. Estimations of the covering functionals of convex hulls of two or more compact convex sets are presented. It is proved that, if a three-dimensional convex body $K$ is the convex hull of two compact convex sets having no interior points, then the least number $c(K)$ of smaller homothetic copies of $K$ needed to cover $K$ is not greater than $8$ and $c(K)=8$ if and only if $K$ is a parallelepiped.
Extending results of Suss and Hadwiger (proved by them for the case of convex bodies and positive ratios), we show that compact (respectively, closed) convex sets in the Euclidean space of dimension n are homothetic provided for any given integer m between 2 and n - 1 (respectively, between 3 and n - 1), the orthogonal projections of the sets on every m-dimensional plane are homothetic, where homothety ratio and its sign may depend on the projection plane. The proof uses a refined version of Straszewicz's theorem on exposed points of compact convex sets.
We present a probabilistic model of illuminating a convex body by independently distributed light sources. In addition to recovering C.A. Rogers' upper bounds for the illumination number, we improve previous estimates of J. Januszewski and M. Naszódi for a generalized version of the illumination parameter.
We show that every $3$-dimensional convex body can be covered by $14$ smaller homothetic copies. The previous result was $16$ copies established by Papadoperakis in 1999, while a conjecture by Hadwiger is $8$. We modify Papadoperakis's approach and develop a discretization technique that reduces the problem to verification of feasibility of a number of linear programs with rational coefficients, which is done with computer assistance using exact arithmetic.
A finite family ${\mathcal B}$ of balls with respect to an arbitrary norm in ${\mathbb R}^d$ ($d\geq 2$) is called a non-separable family if there is no hyperplane disjoint from $\bigcup {\mathcal B}$ that strictly separates some elements of ${\mathcal B}$ from all the other elements of ${\mathcal B}$ in ${\mathbb R}^d$. In this paper we prove that if ${\mathcal B}$ is a non-separable family of balls of radii $r_1, r_2,\ldots , r_n$ ($n\geq 2$) with respect to an arbitrary norm in ${\mathbb R}^d$ ($d\geq 2$), then $\bigcup {\mathcal B}$ can be covered by a ball of radius $\sum_{i=1}^n r_i$. Th…
For two planar convex bodies, $C$ and $D$, consider a packing $S$ of $n$ positive homothets of $C$ contained in $D$. We estimate the total perimeter of the bodies in $S$, denoted ${\rm per}(S)$, in terms of ${\rm per}(D)$ and $n$. When all homothets of $C$ touch the boundary of the container $D$, we show that either ${\rm per}(S)=O(\log n)$ or ${\rm per}(S)=O(1)$, depending on how $C$ and $D$ "fit together," and these bounds are the best possible apart from the constant factors. Specifically, we establish an optimal bound ${\rm per}(S)=O(\log n)$ unless $D$ is a convex polygon and every side o…
We prove a sharp stability result concerning how close homothetic sets attaining near-equality in the Brunn-Minkowski inequality are to being convex. In particular, resolving a conjecture of Figalli and Jerison, we show there are universal constants $C_n,d_n>0$ such that for $A \subset \mathbb{R}^n$ of positive measure, if $|\frac{A+A}{2}\setminus A| \le d_n |A|$, then $|\operatorname{co}(A)\setminus A| \le C_n |\frac{A+A}{2}\setminus A|$ for $\operatorname{co}(A)$ the convex hull of $A$.
At a first glance, the problem of illuminating the boundary of a convex body by external light sources and the problem of covering a convex body by its smaller positive homothetic copies appear to be quite different. They are in fact two sides of the same coin and give rise to one of the important longstanding open problems in discrete geometry, namely, the Illumination Conjecture. In this paper, we survey the activity in the areas of discrete geometry, computational geometry and geometric analysis motivated by this conjecture. Special care is taken to include the recent advances that are not …
In this note, we prove the following conjecture by A. Akopyan and V. Vysotsky: If the convex hull of a planar curve $γ$ covers a planar convex figure $K$, then $\operatorname{length}(γ) \geq \operatorname{per} (K) - \operatorname{diam} (K)$. In addition, all cases of equality in this inequality are studied.
In this paper we deal with problems concerning the volume of the convex hull of two "connecting" bodies. After a historical background we collect some results, methods and open problems, respectively.
N. Dolbilin and M. Tanemura studied the convex hulls of finite subsets of the Clifford torus $T$ in $E^4$. They have completely studied the combinatorial structure of the convex hull for a periodic point set. Moreover, there was performed a numerical simulation of the convex hull for the Poisson point process on $T$ that showed that the mean valence of a vertex of the convex hull has asymptotics $O^*(\ln λ)$ where $λ$ is the rate of the process. N. Dolbilin suggested the author to prove the conjecture on the logarithmic growth of the mean degree of a vertex. In this paper we prove this conject…
In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean $n$-space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are translates, or reflected copies of each other about a common point or a hyperplane containing it. In particular, we give a proof of a related conjecture of Rogers and Shephard.
We study the facial structure and Carathéodory number of the convex hull of an orbit of the group of rotations in R^3 acting on the space of pairs of anisotropic symmetric 3\times 3 tensors. This is motivated by the problem of determining the structure of some proteins in aqueous solution.
In this paper, we provide a lower bound for an area of the convex hull of points and a rectangle in a plane. We then apply this estimate to establish a lower bound for a universal cover problem. We showed that a convex universal cover for a unit length curve has area at least 0.232239. In addition, we show that a convex universal cover for a unit closed curve has area at least 0.0879873.
We describe singularities of the convex hull of a generic compact smooth hypersurface in four-dimensional affine space up to diffeomorphisms. It turns out there are only two new singularities (in comparison with the previous dimension case) which appear at separate points of the boundary of the convex hull and are not removed by a small perturbation of the original hypersurface. The first singularity does not contain functional, but has at least nine continuous number invariants. A normal form which does not contain invariants at all is found for the second singularity.
A generalization of pairwise intersecting Minkowski arrangement of centrally symmetric convex bodies is the pairwise intersecting Minkowski arrangement of order $μ$. Here, the homothetic copies of a centrally symmetric convex body are so that none of their interiors intersect the $μ$-kernel of any other. We give general upper and lower bounds on the cardinality of such arrangements, and study two special cases: For $d$-dimensional translates in classical pairwise intersecting Minkowski arrangement we prove that the sharp upper bound is $3^d$. For $μ=1$ the general version yields to another kno…
We prove that for any convex polygon $S$ with at least four sides, or a concave one with no parallel sides, and any $m>0$, there is an $m$-fold covering of the plane with homothetic copies of $S$ that cannot be decomposed into two coverings.
We describe an algorithm for computing the convex hull of a finite collection of points in the affine building of SL_d(K), for K a field with discrete valuation. These convex hulls describe the relations among a finite collection of invertible matrices over K. As a consequence, we bound the dimension of the tropical projective space needed to realize the convex hull as a tropical polytope.
In this paper, an effective method with time complexity of $\mathcal{O}(K^{3/2}N^2\log \frac{K}{ε_0})$ is introduced to find an approximation of the convex hull for $N$ points in dimension $n$, where $K$ is close to the number of vertices of the approximation. Since the time complexity is independent of dimension, this method is highly suitable for the data in high dimensions. Utilizing a greedy approach, the proposed method attempts to find the best approximate convex hull for a given number of vertices. The approximate convex hull can be a helpful substitute for the exact convex hull for on-…
This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem POLYTOPE-COMPLETENESS-COMBINATORIAL, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the ``no''-case of POLYTOPE-COMPLETENESS-COMBINATORIAL has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).
Let $K \in \R^d$ be a convex body, and assume that $L$ is a randomly rotated and shifted integer lattice. Let $K_L$ be the convex hull of the (random) points $K \cap L$. The mean width $W(K_L)$ of $K_L$ is investigated. The asymptotic order of the mean width difference $W(łK)-W((łK)_L)$ is maximized by the order obtained by polytopes and minimized by the order for smooth convex sets as $ł\to \infty$.
In this paper we focus on the tropical convex hull of convex sets and polyhedral complexes. We give a vertex description of the tropical convex hull of a line segment and a ray. %in \RR^{n+1}/\RR\mathbf{1}. Next we show that tropical convex hull and ordinary convex hull commute in two dimensions and characterize tropically convex polyhedra in any dimension. %$\mathbb{R}^3/\mathbb {R}\mathbf{1}$. Finally we show that the dimension of a tropically convex fan depends on the coordinates of its rays and give a lower bound on the degree of a fan tropical curve using only tropical techniques.
We derive explicit formulae for the expected volume and the expected number of facets of the convex hull of several multidimensional Gaussian random walks in terms of the Gaussian persistence probabilities. Special cases include the already known results about the convex hull of a single Gaussian random walk and the $d$-dimensional Gaussian polytope with or without the origin.
Extending results of Hershberger and Suri for the Euclidean plane, we show that ball hulls and ball intersections of sets of $n$ points in strictly convex normed planes can be constructed in $O(n \log n)$ time. In addition, we confirm that, like in the Euclidean subcase, the $2$-center problem with constrained circles can be solved also for strictly convex normed planes in $O(n^2)$ time. Some ideas for extending these results to more general types of normed planes are also presented.
A rectilinear drawing of a graph $G$ is optimal if it has the smallest number of crossings among all rectilinear drawings of $G$. In this paper it is shown that for $n\geq 8$, the second convex hull of every optimal rectilinear drawing of the complete graph $K_n$ is a triangle.
We show that there exists a universal constant C>0 such that the convex hull of any N points in the hyperbolic space H^n is of volume smaller than C N, and that for any dimension n there exists a constant C_n > 0 such that for any subset A of H^n, Vol(Conv(A_1)) < C_n Vol(A_1) where A_1 is the set of points of hyperbolic distance to A smaller than 1.
Globally non-positively curved, or CAT(0), polyhedral complexes arise in a number of applications, including evolutionary biology and robotics. These spaces have unique shortest paths and are composed of Euclidean polyhedra, yet many algorithms and properties of shortest paths and convex hulls in Euclidean space fail to transfer over. We give an algorithm, using linear programming, to compute the convex hull of a set of points in a 2-dimensional CAT(0) polyhedral complex with a single vertex. We explore the use of shortest path maps to answer single-source shortest path queries in 2-dimensiona…
Three configurations of two perpendicular disks in R^3 are examined, the first in which the disks share centers and the other two in which the disks touch at precisely one point. Volume, surface area and mean width calculations dominate the discussion. Integrated mean curvature also appears as an indirect way to compute mean width.
