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Curvature on a graph via its geometric spectrum

Paul Baird (2013)

Actes des rencontres du CIRM

We approach the problem of defining curvature on a graph by attempting to attach a ‘best-fit polytope’ to each vertex, or more precisely what we refer to as a configured star. How this should be done depends upon the global structure of the graph which is reflected in its geometric spectrum. Mean curvature is the most natural curvature that arises in this context and corresponds to local liftings of the graph into a suitable Euclidean space. We discuss some examples.

Hierarchical models, marginal polytopes, and linear codes

Thomas Kahle, Walter Wenzel, Nihat Ay (2009)

Kybernetika

In this paper, we explore a connection between binary hierarchical models, their marginal polytopes, and codeword polytopes, the convex hulls of linear codes. The class of linear codes that are realizable by hierarchical models is determined. We classify all full dimensional polytopes with the property that their vertices form a linear code and give an algorithm that determines them.

Isometric Embeddings of Pro-Euclidean Spaces

Barry Minemyer (2015)

Analysis and Geometry in Metric Spaces

In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank...

Linear combinations of partitions of unity with restricted supports

Christian Richter (2002)

Studia Mathematica

Given a locally finite open covering of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f: X → E which admit a representation f = ∑ C ∈ a C φ C with a C ∈ E and a partition of unity φ C : C ∈ subordinate to . As an application, we determine the class of all functions f ∈ C(||) on the underlying space || of a Euclidean complex such that, for each polytope P ∈ , the restriction f | P attains its extrema at vertices of P. Finally, a class of extremal functions on the metric space...

Currently displaying 21 – 40 of 85