Related papers: Width Distributions for Convex Regular Polyhedra
The deviation of a general convex body with twice differentiable boundary and an arbitrarily positioned polytope with a given number of vertices is studied. The paper considers the case where the deviation is measured in terms of the…
For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces, by replacing faces with regular spherical polygons in the unit sphere and gluing them edge-to-edge. We consider the class of…
The manuscript provides formulas for the volume of a body defined by the intersection of a solid cone and a solid sphere as a function of the sphere radius, of the distance between cone apex and sphere center, and of the cone aperture…
We compute the p-widths, $\{\omega_p\}$, for the real projective plane with the standard metric.
In this paper, we establish mean width inequalities of sections and projections of convex bodies for isotropic measures with complete equality conditions, which extends the recent work of Alonso-Guti\'{e}rrez and Brazitikos. Different from…
If $C_1$ is the convex hull of the curve of the standard Brownian motion in the complex plane watched from 0 to 1, we consider the convex hulls of $C_1$ and several rotations of it and we compute the mean of the length of their perimeter by…
Let $K\subset\mathbb S^{d-1}$ be a convex spherical body. Denote by $\Delta(K)$ the distance between two random points in $K$ and denote by $\sigma(K)$ the length of a random chord of $K$. We explicitly express the distribution of…
We overview the volume calculations for polyhedra in Euclidean, spherical and hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary tetrahedron in $H^3$ and $S^3$. We also present some results, which provide a…
This article is a survey of recent results on slicing inequalities for convex bodies. The focus is on the setting of arbitrary measures in place of volume.
The largest discs contained in a regular tetrahedron lie in its faces. The proof is closely related to the theorem of Fritz John characterising ellipsoids of maximal volume contained in convex bodies.
We consider random polytopes in the $d$-dimensional Euclidean space that are the convex hulls i.i.d. random points selected according to beta-prime distributions. These distributions are rotationally symmetric, heavy-tailed, and their…
A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main result is explicit formulas for the…
We are generalizing to higher dimensions the Bavard-Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d-dimensional polyhedra with fixed directions of facet normals has a…
This paper derives the exact cumulative density function of the distance between a randomly located node and any arbitrary reference point inside a regular $\el$-sided polygon. Using this result, we obtain the closed-form probability…
This paper studies the straight skeleton of polyhedra in three dimensions. We first address voxel-based polyhedra (polycubes), formed as the union of a collection of cubical (axis-aligned) voxels. We analyze the ways in which the skeleton…
We construct a hollow lattice polytope (resp. a hollow lattice simplex) of dimension $14$ (resp.$~404$) and of width $15$ (resp.$~408$). They are the first known hollow lattice polytopes of width larger than dimension. We also construct a…
The convex hull of N independent random points chosen on the boundary of a simple polytope in R^n is investigated. Asymptotic formulas for the expected number of vertices and facets, and for the expectation of the volume difference are…
The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take…
If we fix the angles at the vertices of a convex planar $n$-gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of $n$ edge lengths form a convex…
In this paper we show that a spherical convex body $C$ is of constant diameter $\tau$ if and only if $C$ is of constant width $\tau$, for $0<\tau<\pi$. Moreover, some applications to Wulff shapes are given.