# 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.

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## 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

• Mathematics
• 1975
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

• Mathematics
• 1977
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