On universally rigid frameworks on the line

A $d$-dimensional bar-and-joint framework $(G,p)$ with underlying graph $G$ is called universally rigid if all realizations of $G$ with the same edge lengths, in all dimensions, are congruent to $(G,p)$. We give a complete characterization of universally rigid one-dimensional bar-and-joint frameworks in general position with a complete bipartite underlying graph. We show that the only bipartite graph for which all generic $d$-dimensional realizations are universally rigid is the complete graph on two vertices, for all $d\geq 1$. We also discuss several open questions concerning generically universally rigid graphs and the universal rigidity of general frameworks on the line.

Ball hulls, ball intersections, and 2-center problems for gauges

The notions of ball hull and ball intersection of nite sets, important in Banach space theory, are extended from normed planes to generalized normed planes, i.e., to (asymmetric) convex distance functions which are also called gauges. In this more general setting we derive various new results about these notions and their relations to each other. Further on, we extend the known 2-center problem and a modified version of it from the Euclidean situation to norms and gauges or, in other words, from Euclidean circles to arbitrary closed convex curves. We derive algorithmical results on the construction of ball hulls and ball intersections, and computational approaches to the 2-center problem with constrained circles and, in case of strictly convex norms and gauges, for the fixed 2-center problem are also given

On affine rigidity

We define the notion of affine rigidity of a hypergraph and prove a variety of fundamental results for this notion. First, we show that affine rigidity can be determined by the rank of a specific matrix which implies that affine rigidity is a generic property of the hypergraph.Then we prove that if a graph is is $(d+1)$-vertex-connected, then it must be "generically neighborhood affinely rigid" in $d$-dimensional space. This implies that if a graph is $(d+1)$-vertex-connected then any generic framework of its squared graph must be universally rigid. Our results, and affine rigidity more generally, have natural applications in point registration and localization, as well as connections to manifold learning.Comment: Updated abstrac

Convex Geometry and its Applications

The past 30 years have not only seen substantial progress and lively activity in various areas within convex geometry, e.g., in asymptotic geometric analysis, valuation theory, the $L_p$-Brunn-Minkowski theory and stochastic geometry, but also an increasing amount and variety of applications of convex geometry to other branches of mathematics (and beyond), e.g. to PDEs, statistics, discrete geometry, optimization, or geometric algorithms in computer science. Thus convex geometry is a flourishing and attractive field, which is also reflected by the considerable number of talented young mathematicians at this meeting

Characterizing the universal rigidity of generic frameworks

A framework is a graph and a map from its vertices to E^d (for some d). A framework is universally rigid if any framework in any dimension with the same graph and edge lengths is a Euclidean image of it. We show that a generic universally rigid framework has a positive semi-definite stress matrix of maximal rank. Connelly showed that the existence of such a positive semi-definite stress matrix is sufficient for universal rigidity, so this provides a characterization of universal rigidity for generic frameworks. We also extend our argument to give a new result on the genericity of strict complementarity in semidefinite programming.Comment: 18 pages, v2: updates throughout; v3: published versio

Discrete Differential Geometry

This is the collection of extended abstracts for the 26 lectures and the open problems session at the second Oberwolfach workshop on Discrete Differential Geometry