Related papers: How to decide whether two convex octahedra are aff…
We study the limiting case of the Krichever construction of orthogonal curvilinear coordinate systems when the spectral curve becomes singular. We show that the case when the curve is reducible and all its irreducible components are…
We present structures comprised of identical convex polyhedra which are interlocked geometrically. These sets cannot be disassembled by removing individual polyhedra by translations and/or rotations. The shapes that permit interlocking…
Coherent states for equally spaced, homogeneous waveguide arrays are defined, in the infinite, semiinfinite and finite cases, and resolutions of the identity are constructed, using different methods. In the infinite case, which corresponds…
We study the Carath\'eodory number of homogeneous convex cones via their spectrahedral representations. A characterization of homogeneous convex cones whose ranks match their Carath\'eodory numbers is given. This characterization is then…
We conjecture that a convex polytope is uniquely determined up to isometry by its edge-graph, edge lengths and the collection of distances of its vertices to some arbitrary interior point, across all dimensions and all combinatorial types.…
We consider point sets in the $m$-dimensional affine space $\mathbb{F}_q^m$ where each squared Euclidean distance of two points is a square in $\mathbb{F}_q$. It turns out that the situation in $\mathbb{F}_q^m$ is rather similar to the one…
Let $F$ be Cayley's ruled cubic surface in a projective three-space over any commutative field $K$. We determine all collineations fixing $F$, as a set, and all cubic forms defining $F$. For both problems the cases $|K|=2,3$ turn out to be…
We present a complete system of inequalities for the inradius, circumradius, and diameter in the $3$-dimensional Euclidean space. To do so, we prove quasiconcavity of the inradius evaluated over $n$-simplices with a common facet…
We prove two results about transforming any convex polyhedron, modeled as a linkage L of its edges. First, if we subdivide each edge of L in half, then L can be continuously flattened into a plane. Second, if L is equilateral and we again…
This paper investigates sufficient and necessary conditions for the existence of a homotopy equivalence between two finite simplicial complexes from an algorithmic point of view. As a result, the conditions are formulated in terms of the…
We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have…
In this note we establish some appropriate conditions for stochastic equality of two random variables/vectors which are ordered with respect to convex ordering or with respect to supermodular ordering. Multivariate extensions of this result…
This paper is devoted to the study the $m$-point homogeneity property and the point homogeneity degree for finite metric spaces. Since the vertex sets of regular polytopes, as well as of some their generalizations, are homogeneous, we pay…
Convex geometries (Edelman and Jamison, 1985) are finite combinatorial structures dual to union-closed antimatroids or learning spaces. We define an operation of resolution for convex geometries, which replaces each element of a base convex…
Optimizing the free energy under a mass constraint may generate a convex crystal subject to assumptions on the potential $g(0)=0$, $g \ge 0$. The general problem classically attributed to Almgren is to infer if this is the case assuming the…
We study N=2 superconformal theories on Euclidean and Lorentzian four-manifolds with a view toward applications to holography and localization. The conditions for supersymmetry are equivalent to a set of differential constraints including a…
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…
It is proved that every convex body in the plane has a point such that the union of the body and its image under reflection in the point is convex. If the body is not centrally symmetric, then it has, in fact, three affinely independent…
We focus on a branch of region-based spatial logics dealing with affine geometry. The research on this topic is scarce: only a handful of papers investigate such systems, mostly in the case of the real plane. Our long-term goal is to…
I give a characterization of the conditions for two Hamiltonians to be equivalent, discuss the construction of the operators that relate equivalent Hamiltonians, and introduce variational methods that can select Hamiltonians with desirable…