Diameter-extremal subsets of spheres

@article{Katz1989DiameterextremalSO,
  title={Diameter-extremal subsets of spheres},
  author={Mikhail G. Katz},
  journal={Discrete \& Computational Geometry},
  year={1989},
  volume={4},
  pages={117-137}
}
  • M. Katz
  • Published 1 December 1989
  • Mathematics
  • Discrete & Computational Geometry
We investigate those spherical point sets which, relative to the Hausdorff metric, give local minima of the diameter function, and obtain estimates which, in principle, justify computer-generated configurations. We test the new sets against two classical conjectures: Borsuk's and McMullen's. 

Counterexample to a geodesic length conjecture on the 2-sphere

We describe a convex surface of constant width in R 3 with tetrahedral symmetry, and show that it provides a counterexample to a conjectured inequality L ≤ 2D relating the intrinsic diameter D and

Convexity, critical points, and connectivity radius

  • M. Katz
  • Mathematics
    Proceedings of the American Mathematical Society
  • 2019
We study the level sets of the distance function from a boundary point of a convex set in Euclidean space. We provide a lower bound for the range of connectivity of the level sets, in terms of the

Filling area conjecture, optimal systolic inequalities, and the fiber class in abelian covers

We investigate the filling area conjecture, optimal systolic inequalities, and the related problem of the nonvanishing of certain linking numbers in 3-manifolds.

Pyramids in the complex projective plane

We study the critical points of the diameter functional δ on the n-fold Cartesian product of the complex projective plane CP2 with the Fubini-Study metric. Such critical points arise in the

Geometric Approaches on Persistent Homology

TLDR
Several geometric notions, including thick-thin decompositions and the width of a homology class, are introduced to the theory of persistent homology, and quantitative and geometric descriptions of the “size” or “persistence” of ahomology class are given.

Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius

TLDR
A different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space into a larger space and then considering thickenings of the original space is considered.

Vietoris thickenings and complexes have isomorphic homotopy groups

. We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let U be a cover of a separable metric space X by open sets with a uniform

The Persistent Topology of Optimal Transport Based Metric Thickenings

A metric thickening of a given metric space X is any metric space admitting an isometric embedding of X . Thickenings have found use in applications of topology to data analysis, where one may

On Vietoris-Rips complexes of hypercube graphs

We describe the homotopy types of Vietoris–Rips complexes of hypercube graphs at small scale parameters. In more detail, let Qn be the vertex set of the hypercube graph with 2n vertices, equipped

References

SHOWING 1-10 OF 14 REFERENCES

Curvature, diameter and betti numbers

We give an upper bound for the Betti numbers of a compact Riemannian manifold in terms of its diameter and the lower bound of the sectional curvatures. This estimate in particular shows that most

The numbers of faces of simplicial polytopes

In this paper is considered the problem of determining the possiblef-vectors of simplicial polytopes. A conjecture is made about the form of the sclution to this problem; it is proved in the case

Comparison theorems in Riemannian geometry

Basic concepts and results Toponogov's theorem Homogeneous spaces Morse theory Closed geodesics and the cut locus The sphere theorem and its generalizations The differentiable sphere theorem Complete

The filling radius of two-point homogeneous spaces

is an isometric imbedding, as dist(;c, x') — \\dx — dx,\\ (the triangle inequality). Consider the inclusion homomorphism aε: Hn(X) -* Hn(UεX% where UεX C L^iX) is the ε-neighborhood of X, and the

The lower bound conjecture for 3- and 4-manifolds

For any closed connected d-manifold M let ](M) denote the set of vectors / (K)= (]0(K) ..... fd(K)), where K ranges over all triangulations of M and ]k(K) denotes the number of k-simplices of K. The

A generalized sphere theorem

A basic problem in Riemannian geometry is the study of relations between the topological structure and the Riemannian structure of a complete, connected Riemannian manifold M of dimension n > 2. By a

A Proof of the Sufficiency of McMullen's Conditions for f-Vectors of Simplicial Convex Polytopes

The number of faces of a simplicial convex polytope

Convex Polytopes and the Upper Bound Conjecture