1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

/pdf/convex-bodies-the-brunn-minkowski-theory-3raynvod45.pdf

Convex bodies : the Brunn-Minkowski theory

A scheme is developed for classifying the types of motion perceived by a humanlike robot. It is assumed that the robot receives visual images of the scene using a perspective system model. Equations, theorems, concepts, clues, etc., relating the objects, their positions, and their motion to their images on the focal plane are presented. >

http://ieeexplore.ieee.org/iel2/815/3148/00100651.pdf

Robot vision

Convex bodies: the brunn–minkowski theory

In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered includes those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include well-studied cases from many technical fields such as sparse vectors (signal processing, statistics) and low-rank matrices (control, statistics), as well as several others including sums of a few permutation matrices (ranked elections, multiobject tracking), low-rank tensors (computer vision, neuroscience), orthogonal matrices (machine learning), and atomic measures (system identification). The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial structure of the atomic norm ball carries a number of favorable properties that are useful for recovering simple models, and an analysis of the underlying convex geometry provides sharp estimates of the number of generic measurements required for exact and robust recovery of models from partial information. These estimates are based on computing the Gaussian widths of tangent cones to the atomic norm ball. When the atomic set has algebraic structure the resulting optimization problems can be solved or approximated via semidefinite programming. The quality of these approximations affects the number of measurements required for recovery, and this tradeoff is characterized via some examples. Thus this work extends the catalog of simple models (beyond sparse vectors and low-rank matrices) that can be recovered from limited linear information via tractable convex programming.

/pdf/the-convex-geometry-of-linear-inverse-problems-22nrjt46yu.pdf

The Convex Geometry of Linear Inverse Problems

Lectures on Cauchy’s Problem in Linear Partial Differential Equations. By J. Hadamard. Pp. viii+316. 15s.net. 1923. (Per Oxford University Press.)

This paper deals with a problem from computational convexity and its application to computer algebra. This paper determines the complexity of computing the Minkowski sum of k convex polytopes in $\mathbb{R}^d $, which are presented either in terms of vertices or in terms of facets. In particular, if the dimension d is fixed, the authors obtain a polynomial time algorithm for adding k polytopes with up to n vertices. The second part of this paper introduces dynamic versions of Buchberger’s Grobner bases algorithm for polynomial ideals. Using the Minkowski addition of Newton polytopes, the authors show that the following problem can be solved in polynomial time for any finite set of polynomials $\mathcal{T} \subset K [ x_1 , \ldots ,x_d ]$, where d is fixed: Does there exist a term order $\tau $ such that $\mathcal{T}$ is a Grobner basis for its ideal with respect to $\tau $? If not, find an optimal term order for $\mathcal{T}$ with respect to a natural Hilbert function criterion.

Minkowski addition of polytopes: computational complexity and applications to Gro¨bner bases

We study the determination of finite subsets of the integer lattice En, n > 2, by X-rays. In this context, an X-ray of a set in a direction u gives the number of points in the set on each line parallel to u. For practical reasons, only X-rays in lattice directions, that is, directions parallel to a nonzero vector in the lattice, are permitted. By combining methods from algebraic number theory and convexity, we prove that there are four prescribed lattice directions such that convex subsets of En (i.e., finite subsets F with F = En n conv F) are determined, among all such sets, by their X-rays in these directions. We also show that three X-rays do not suffice for this purpose. This answers a question of Larry Shepp, and yields a stability result related to Hammer's X-ray problem. We further show that any set of seven prescribed mutually nonparallel lattice directions in 22 have the property that convex subsets of 22 are determined, among all such sets, by their X-rays in these directions. We also consider the use of orthogonal projections in the interactive technique of successive determination, in which the information from previous projections can be used in deciding the direction for the next projection. We obtain results for finite subsets of the integer lattice and also for arbitrary finite subsets of Euclidean space which are the best possible with respect to the numbers of projections used.

/pdf/discrete-tomography-determination-of-finite-sets-by-x-rays-3ldjrwrm99.pdf

Discrete tomography: Determination of finite sets by X-rays

On the computational complexity of reconstructing lattice sets from their x-rays

Note: Professor Pach's number: [065] Reference DCG-ARTICLE-2008-010doi:102307/2323956 Record created on 2008-11-17, modified on 2017-05-12

Peter Gritzmann

Papers

Minkowski addition of polytopes: computational complexity and applications to Gro¨bner bases

Discrete tomography: Determination of finite sets by X-rays

On the computational complexity of reconstructing lattice sets from their x-rays

Embedding a planar triangulation with vertices at specified points

Applied geometry and discrete mathematics : the Victor Klee festschrift