Related papers: Convex hulls of polyominoes

We determine the maximal hyperplane sections of the regular $n$-simplex, if the distance of the hyperplane to the centroid is fairly large, i.e. larger than the distance of the centroid to the midpoint of edges. Similar results for the…

Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex polytopes exactly equals the normalized…

We state a conjecture about the volume of symplectically self-polar convex bodies and show that it is equivalent to Mahler's conjecture concerning the volume of a convex body and its Euclidean polar. We also establish lower and upper bounds…

A flexible polyhedron in an n-dimensional space of constant curvature, namely, in the Euclidean space, or in the Lobachevsky space, or in the sphere, is a polyhedron with rigid (n-1)-dimensional faces and hinges at (n-2)-dimensional faces.…

We prove the following sparse approximation result for polytopes. Assume that $Q$ is a polytope in John's position. Then there exist at most $2d$ vertices of $Q$ whose convex hull $Q'$ satisfies $Q \subseteq - 2d^2 \, Q'$. As a consequence,…

Consider two half-spaces $H_1^+$ and $H_2^+$ in $\mathbb{R}^{d+1}$ whose bounding hyperplanes $H_1$ and $H_2$ are orthogonal and pass through the origin. The intersection $\mathbb{S}_{2,+}^d:=\mathbb{S}^d\cap H_1^+\cap H_2^+$ is a spherical…

Facets of the convex hull of $n$ independent random vectors chosen uniformly at random from the unit sphere in $\mathbb{R}^d$ are studied. A particular focus is given on the height of the facets as well as the expected number of facets as…

Since $n$-dimensional $\lambda$-hypersurfaces in the Euclidean space $\mathbb {R}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of…

We study extremal properties of spherical random polytopes, the convex hull of random points chosen from the unit Euclidean sphere in $\mathbb{R}^n$. The extremal properties of interest are the expected values of the maximum and minimum…

We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$…

For a given $\lambda >0$, a convex body in $\mathbb R^n$ is $\lambda$-convex if it is the intersection of (finitely or infinitely many) balls of radius $1/\lambda$. In this note, we show that among all $\lambda$-convex bodies in $\mathbb…

Up to a factor 1/n!, the volume of a big line bundle agrees with the Euclidean volume of its Okounkov body. The latter is the convex hull of top rank valuation vectors of sections, all with respect to a single flag. In this text we give a…

This study presents a novel algorithm for identifying the set of extreme points that constitute the exact convex hull of a point set in high-dimensional Euclidean space. The proposed method iteratively solves a sequence of dynamically…

The convex hulls of face-vertex incident vectors of 3-face-colorable convex polytopes are computed. It is found that every such convex hull is a $d$-polytope with $d+2$ or $d+3$ vertices. Utilizing Gale transform and Gale diagram, we…

This article is concerned with the problem of placing seven or eight points on the unit sphere $\mathbb{S}^2$ in $\mathbb{R}^3$ so that the surface area of the convex hull of the points is maximized. In each case, the solution is given for…

A subset of the Hamming cube over $n$-letter alphabet is said to be $d$-maximal if its diameter is $d$, and adding any point increases the diameter. Our main result shows that each $d$-maximal set is either of size at most $(n+o(n))^d$ or…

The bellows conjecture claims that the volume of any flexible polyhedron of dimension 3 or higher is constant during the flexion. The bellows conjecture was proved for flexible polyhedra in the Euclidean spaces of dimensions 3 and higher,…

Extremal elements and a h-hull of sets in the n-dimensional hypercomplex space are investigated. Introduced a class of H-quasiconvex sets including strongly hypercomplex convex sets and being closed with respect to intersections.

We prove bounds for the volume of neighborhoods of algebraic sets, in the euclidean space or the sphere, in terms of the degree of the defining polynomials, the number of variables and the dimension of the algebraic set, without any…

We show that if M is a complete, finite-volume, hyperbolic 3-manifold having exactly one cusp, and if H_1(M;Z_2) has dimension at least 6, then M has volume greater than 5.06. We also show that if M is a closed, orientable hyperbolic…