Related papers: Additive structure in convex translates

We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann's theorem to the complex case by proving that complex…

Convex polyhedra are the basis for several abstractions used in static analysis and computer-aided verification of complex and sometimes mission critical systems. For such applications, the identification of an appropriate…

This tutorial provides an exposition of a flexible geometric framework for high dimensional estimation problems with constraints. The tutorial develops geometric intuition about high dimensional sets, justifies it with some results of…

We extend many theorems from the context of solid angle sums over rational polytopes to the context of solid angle sums over real polytopes. Moreover, we consider any real dilation parameter, as opposed to the traditional integer dilation…

Filliman duality expresses (the characteristic measure of) a convex polytope P containing the origin as an alternating sum of simplices that share supporting hyperplanes with P. The terms in the alternating sum are given by a triangulation…

The unit vectors transformation between the Cartesian and the novel Similar Oblate Spheroidal coordinates, and vice versa, is derived. It can help to transform vector fields between these two types of orthogonal coordinates which can…

Convergence theory is an extension of general topology. In contrast with topology, it is closed under some important operations, like exponentiation. With all its advantages, convergence theory remains rather unknown. It is an aim of this…

Let $P$ and $Q$ be two simple polygons in the plane of total complexity $n$, each of which can be decomposed into at most $k$ convex parts. We present an $(1-\varepsilon)$-approximation algorithm, for finding the translation of $Q$, which…

We define and analyze various generalizations of the punctual Hilbert scheme of the plane, associated to complex or real Lie algebras. Out of these, we construct new geometric structures on surfaces whose moduli spaces share multiple…

We prove, in all dimensions $n\geq 2$, that there exists a convex translator lying in a slab of width $\pi\sec\theta$ in $\mathbb{R}^{n+1}$ (and in no smaller slab) if and only if $\theta\in[0,\frac{\pi}{2}]$. We also obtain convexity and…

Spurious ambiguity is the phenomenon whereby distinct derivations in grammar may assign the same structural reading, resulting in redundancy in the parse search space and inefficiency in parsing. Understanding the problem depends on…

We prove addition and subspace theorems for asymptotic large inductive dimension. We investigate a transfinite extension of this dimension and show that it is trivial.

Let $S$ be a set of $n$ points in general position in the plane, and let $X_{k,\ell}(S)$ be the number of convex $k$-gons with vertices in $S$ that have exactly $\ell$ points of $S$ in their interior. We prove several equalities for the…

Let $S_1$ and $S_2$ be disjoint finite unions of parallelepipeds. We describe necessary and sufficient conditions on the sets $S_1,S_2$ and exponents $p$ such that the canonical projection $P$ from $PW_{S_1\cup S_2}^p$ to $PW_{S_1}^p$ is a…

If the complement of a closed convex set in a closed convex cone is bounded, then this complement minus the apex of the cone is called a coconvex set. Coconvex sets appear in singularity theory (they are closely related to Newton diagrams)…

Using combinatorial methods, we derive several formulas for the volume of convex bodies obtained by intersecting a unit hypercube with a halfspace, or with a hyperplane of codimension 1, or with a flat defined by two parallel hyperplanes.…

We investigate metric projections and distance functions referring to convex bodies in finite-dimensional normed spaces. For this purpose we identify the vector space with its dual space by using, instead of the usual identification via the…

A planar set $P$ is said to be cover-decomposable if there is a constant $k=k(P)$ such that every $k$-fold covering of the plane with translates of $P$ can be decomposed into two coverings. It is known that open convex polygons are…

Let $\mathcal{S}$ be a finite set of integer points in $\mathbb{R}^d$, which we assume has many symmetries, and let $P\in\mathbb{R}^d$ be a fixed point. We calculate the distances from $P$ to the points in $\mathcal{S}$ and compare the…

A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial…