Translational tilings by a polytope, with multiplicity

We study the problem of covering R^d by overlapping translates of a convex body P, such that almost every point of R^d is covered exactly k times. Such a covering of Euclidean space by translations is called a k-tiling. The investigation of tilings (i.e. 1-tilings in this context) by translations began with the work of Fedorov and Minkowski. Here we extend the investigations of Minkowski to k-tilings by proving that if a convex body k-tiles R^d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski's conditions for 1-tiling polytopes. Conversely, in the case that P is a rational polytope, we also prove that if P is centrally symmetric and has centrally symmetric facets, then P must k-tile R^d for some positive integer k

The inverse moment problem for convex polytopes: implementation aspects

We give a detailed technical report on the implementation of the algorithm presented in Gravin et al. (Discrete & Computational Geometry'12) for reconstructing an $N$-vertex convex polytope $P$ in $\mathbb{R}^d$ from the knowledge of $O(Nd)$ of its moments

The inverse moment problem for convex polytopes

The goal of this paper is to present a general and novel approach for the reconstruction of any convex d-dimensional polytope P, from knowledge of its moments. In particular, we show that the vertices of an N-vertex polytope in R^d can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to an unknown polynomial measure od degree D) in d+1 distinct generic directions. Our approach is based on the collection of moment formulas due to Brion, Lawrence, Khovanskii-Pukhikov, and Barvinok that arise in the discrete geometry of polytopes, and what variously known as Prony's method, or Vandermonde factorization of finite rank Hankel matrices.Comment: LaTeX2e, 24 pages including 1 appendi

Commitment, Risk, and Consumption: Do Birds of a

We show that incorporating consumption commitments into a standard model of precautionary saving can complicate the usual relationship between risk and consumption. In particular, the presence of plausible adjustment costs can cause a mean-preserving increase in unemployment risk to lead to increased consumption. The predictions of this model are consistent with empirical evidence from dual-earning couples. Couples who share an occupation face increased risk as their unemployment shocks are more highly correlated. Such couples spend more on owner-occupied housing than other couples, spend no more on rent, and are more likely to rent than own. This pattern is strongest when the household faces higher moving costs, or when unemployment insurance provides a less generous safety net