Related papers: The Intermediate Logic of Convex Polyhedra
We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic…
A spectrahedron is a set defined by a linear matrix inequality. A projection of a spectrahedron is often called a semidefinitely representable set. We show that the convex hull of a finite union of such projections is again a projection of…
While intersections of convex sets are convex, their unions have rather complicated behavior. Some natural contexts where they appear include duality arguments involving boundaries of convex sets and valuations, which have an Euler…
Let $P$ be a (non necessarily convex) embedded polyhedron in $\R^3$, with its vertices on an ellipsoid. Suppose that the interior of $P$ can be decomposed into convex polytopes without adding any vertex. Then $P$ is infinitesimally rigid.…
We define a new logic-induced notion of bisimulation (called $\rho$-bisimulation) for coalgebraic modal logics given by a logical connection, and investigate its properties. We show that it is structural in the sense that it is defined only…
This paper continues the investigation of the logic of competing theories, be they scientific, social, political etc. We introduce a many-valued, multi-type modal language which we endow with relational semantics based on enriched reflexive…
2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level…
In this paper, we develop a new index theory for manifolds with polyhedral boundary. As an application, we prove Gromov's dihedral extremality conjecture regarding comparisons of scalar curvatures, mean curvatures and dihedral angles…
This work provides two sufficient conditions in terms of sections or projections for a convex body to be a polytope. These conditions are necessary as well.
A convex set with nonempty interior is maximal lattice-free if it is inclusion-maximal with respect to the property of not containing integer points in its interior. Maximal lattice-free convex sets are known to be polyhedra. The precision…
The concept of representing a polytope that is associated with some combinatorial optimization problem as a linear projection of a higher-dimensional polyhedron has recently received increasing attention. In this paper (written for the…
We study the equational theory of the Weihrauch lattice with multiplication, meaning the collection of equations between terms built from variables, the lattice operations $\sqcup$, $\sqcap$, the product $\times$, and the finite…
We propose a new contact relation between polytopes. Intuitively, we say that two polytopes are in strong contact if a small enough object can pass from one of them to the other while remaining in their union. In the first half of the paper…
Frames provide redundant, stable representations of data which have important applications in signal processing. We introduce a connection between symplectic geometry and frame theory and show that many important classes of frames have…
We study multiple tilings of 3-dimensional Euclidean space by a convex body. In a multiple tiling, a convex body $P$ is translated with a discrete multiset $\Lambda$ in such a way that each point of the space gets covered exactly $k$ times,…
We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for…
We show that every convex code realizable by compact sets in the plane admits a realization consisting of polygons, and analogously every open convex code in the plane can be realized by interiors of polygons. We give factorial-type bounds…
We introduce a logic to express structural properties of automata with string inputs and, possibly, outputs in some monoid. In this logic, the set of predicates talking about the output values is parametric, and we provide sufficient…
In this Letter we suggest a method of convex rigid frames in the studies of the multipartite quNit pure-states. We illustrate what are the convex rigid frames and what is the method of convex rigid frames. As the applications we use this…
At the heart of convex geometry lies the observation that the volume of convex bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms of the coefficients of this polynomial, called mixed volumes. Among the…