Intrinsic volumes of inscribed random polytopes in smooth convex bodies

Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the $s$-th intrinsic volumes $V_s(K_n)$ of $K_n$ for $s\in\{1, ..., d\}$. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of $K_n$. The essential tools are the Economic Cap Covering Theorem of B\'ar\'any and Larman, and the Efron-Stein jackknife inequality

Querying Visible and Invisible Information

We provide a wide-ranging study of the scenario where a subset of the relations in the schema are visible - that is, their complete contents are known - while the remaining relations are invisible. We also have integrity constraints (invariants given by logical sentences) which may relate the visible relations to the invisible ones. We want to determine which information about a query (a positive existential sentence) can be inferred from the visible instance and the constraints. We consider both positive and negative query information, that is, whether the query or its negation holds. We consider the instance-level version of the problem, where both the query and the visible instance are given, as well as the schema-level version, where we want to know whether truth or falsity of the query can be inferred in some instance of the schema

Equipartitioning by a convex 3-fan

We show that for a given planar convex set K of positive area there exist three pairwise internally disjoint convex sets whose union is K such that they have equal area and equal perimeter

Fairly Allocating Contiguous Blocks of Indivisible Items

In this paper, we study the classic problem of fairly allocating indivisible items with the extra feature that the items lie on a line. Our goal is to find a fair allocation that is contiguous, meaning that the bundle of each agent forms a contiguous block on the line. While allocations satisfying the classical fairness notions of proportionality, envy-freeness, and equitability are not guaranteed to exist even without the contiguity requirement, we show the existence of contiguous allocations satisfying approximate versions of these notions that do not degrade as the number of agents or items increases. We also study the efficiency loss of contiguous allocations due to fairness constraints.Comment: Appears in the 10th International Symposium on Algorithmic Game Theory (SAGT), 201

Algorithms for Colourful Simplicial Depth and Medians in the Plane

The colourful simplicial depth of a point x in the plane relative to a configuration of n points in k colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point x, and of finding a point, called a median, that maximizes colourful simplicial depth. For computing the colourful simplicial depth of x, our algorithm runs in time O(n log(n) + k n) in general, and O(kn) if the points are sorted around x. For finding the colourful median, we get a time of O(n^4). For comparison, the running times of the best known algorithm for the monochrome version of these problems are O(n log(n)) in general, improving to O(n) if the points are sorted around x for monochrome depth, and O(n^4) for finding a monochrome median.Comment: 17 pages, 8 figure

Tverberg’s theorem at 50: Extensions and counterexamples

We describe how a powerful new “constraint method” yields many different extensions of the topological version of Tverberg’s 1966 Theorem in the prime power case— and how the same method also was instrumental in the recent spectacular construction of counterexamples for the general case. © 2016. All rights reserved

Quantum Computing with NMR

A review of progress in NMR quantum computing and a brief survey of the literatureComment: Commissioned by Progress in NMR Spectroscopy (95 pages, no figures

Continuum Surface Energy from a Lattice Model

We investigate connections between the continuum and atomistic descriptions of deformable crystals, using certain interesting results from number theory. The energy of a deformed crystal is calculated in the context of a lattice model with general binary interactions in two dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. The main contribution is an explicit formula for the surface energy density as a function of the deformation gradient and boundary normal. The result is valid for a large class of domains, including faceted (polygonal) shapes and regions with piecewise smooth boundaries.Comment: V. 1: 10 pages, no fig's. V 2: 23 pages, no figures. Misprints corrected. Section 3 added, (new results). Intro expanded, refs added.V 3: 26 pages. Abstract changed. Section 2 split into 2. Section (4) added material. V 4, 28 pages, Intro rewritten. Changes in Sec.5 (presentation only). Refs added.V 5,intro changed V.6 address reviewer's comment

Few smooth d-polytopes with n lattice points

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result