Mathematics

Let k be at most 0, and let X be a locally-finite CAT(k) polyhedral 2-complex X, each face with constant curvature k. Let E be a closed, rectifiably-connected subset of X with trivial first singular homology. We show that E, under the induced path metric, is a complete CAT(k) space.

This paper proposes a closed-form parametric formula of the Minkowski sum boundary for broad classes of convex bodies in d-dimensional Euclidean space. With positive sectional curvatures at every point, the boundary that encloses each body can be characterized by the surface gradient. The first theorem directly parameterizes the Minkowski sums using the unit normal vector at each body surface. Although simple to express mathematically, such a parameterization is not always practical to obtain computationally. Therefore, the second theorem derives a more useful parametric closed-form expression using the gradient that is not normalized. In the special case of two ellipsoids, the proposed expressions are identical to those derived previously using geometric interpretations. In order to further examine the results, numerical verifications and comparisons of the Minkowski sums between two superquadric bodies are conducted. The application for the generation of configuration space obstacles in motion planning problems is introduced and demonstrated.

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of m arbitrary ellipsoids in N -dimensional Euclidean space. Expressions for the principal curvatures of these Minkowski sums are also derived. These results are then used to obtain upper and lower volume bounds for the Minkowski sum of ellipsoids in terms of their defining matrices; the lower bounds are sharper than the Brunn-Minkowski inequality. A reverse isometric inequality for convex bodies is also given.

A coarse compactification of a proper metric space X is any compactification of X that is dominated by its Higson compactification. In this paper we describe the maximal coarse compactification of X whose corona is of dimension 0 . In case of geodesic spaces X , it coincides with the Freundenthal compactification of X . As an application we provide an alternative way of extending the concept of the number of ends from finitely generated groups to arbitrary countable groups. We present a geometric proof of a generalization of Stallings' theorem by showing that any countable group of two ends contains an infinite cyclic subgroup of finite index. Finally, we define ends of arbitrary coarse spaces.

In the present paper, we introduce a concept of Ricci curvature on hypergraphs for a nonlinear Laplacian. We prove that our definition of the Ricci curvature is a generalization of Lin-Lu-Yau coarse Ricci curvature for graphs to hypergraphs. We also show a lower bound of nonzero eigenvalues of Laplacian, gradient estimate of heat flow, and diameter bound of Bonnet-Myers type for our curvature notion. This research leads to understanding how nonlinearity of Laplacian causes complexity of curvatures.

In this paper we provide several \emph{metric universality} results. We exhibit for certain classes $\cC$ of metric spaces, families of metric spaces ( M i , d i ) i∈I which have the property that a metric space (X, d X ) in $\cC$ is coarsely, resp. Lipschitzly, universal for all spaces in $\cC$ if the collection of spaces ( M i , d i ) i∈I equi-coarsely, respectively equi-Lipschitzly, embeds into (X, d X ) . Such families are built as certain Schreier-type metric subsets of $\co$. We deduce a metric analog to Bourgain's theorem, which generalized Szlenk's theorem, and prove that a space which is coarsely universal for all separable reflexive asymptotic- c 0 Banach spaces is coarsely universal for all separable metric spaces. One of our coarse universality results is valid under Martin's Axiom and the negation of the Continuum Hypothesis. We discuss the strength of the universality statements that can be obtained without these additional set theoretic assumptions. In the second part of the paper, we study universality properties of Kalton's interlacing graphs. In particular, we prove that every finite metric space embeds almost isometrically in some interlacing graph of large enough diameter.

This paper studies coarse compactifications and their boundary. We introduce two alternative descriptions to Roe's original definition of coarse compactification. One approach uses bounded functions on X that can be extended to the boundary. They satisfy the Higson property exactly when the compactification is coarse. The other approach defines a relation on subsets of X which tells when two subsets closure meet on the boundary. A set of axioms characterizes when this relation defines a coarse compactification. Such a relation is called large-scale proximity. Based on this foundational work we study examples for coarse compactifications Higson compactification, Freudenthal compactification and Gromov compactification. For each example we characterize the bounded functions which can be extended to the coarse compactification and the corresponding large-scale proximity relation. We provide an alternative proof for the property that the Higson compactification is universal among coarse compactifications. Furthermore the Freudenthal compactification is universal among coarse compactifications with totally disconnected boundary. If X is hyperbolic geodesic proper then there is a closed embedding ν( R + )×∂X→ν(X) . Its image is a retract of ν(X) if X is a tree.

We introduce the notion of coarse metric. Every coarse metric induces a coarse structure on the underlying set. Conversely, we observe that all coarse spaces come from a particular type of coarse metric in a unique way. In the case when the coarse structure E on a set X is defined by a coarse metric that takes values in a meet-complete totally ordered set, we define the associated Hausdorff coarse metric on the set P 0 (X) of non-empty subsets of X and show that it induces the Hausdorff coarse structure on P 0 (X) . On the other hand, we define the notion of pseudo uniform metric. Each pseudo uniform metric induces a uniform structure on the underlying space. In the reverse direction, we show that a uniform structure U on a set X is induced by a map d from X×X to a partially ordered set (with no requirement on d ) if and only if U admits a base B such that B∪{⋂U} is closed under arbitrary intersections. In this case, U is actually defined by a pseudo uniform metric. We also show that a uniform structures U comes from a pseudo uniform metric that takes values in a totally ordered set if and only if U admits a totally ordered base. Finally, a valuation ring will produce an example of a coarse and pseudo uniform metric that take values in a totally ordered set.

We prove a "multiple colored Tverberg theorem" and a "balanced colored Tverberg theorem", by applying different methods, tools and ideas. The proof of the first theorem uses multiple chessboard complexes (as configuration spaces) and Eilenberg-Krasnoselskii theory of degrees of equivariant maps for non-free actions. The proof of the second result relies on high connectivity of the configuration space, established by discrete Morse theory.

We prove the following Helly-type result. Let C 1 ,…, C 3d be finite families of convex bodies in R d . Assume that for any colorful selection of 2d sets, C i k ∈ C i k for each 1≤k≤2d with 1≤ i 1 <⋯< i 2d ≤3d , the intersection ⋂ k=1 2d C i k is of volume at least 1. Then there is an 1≤i≤3d such that ⋂ C∈ C i C is of volume at least d −O( d 2 ) .