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The Minimal Spherical Dispersion.

Joscha Prochno1, Daniel Rudolf2

  • 1Faculty of Computer Science and Mathematics, University of Passau, Dr.-Hans-Kapfinger-Straße 30, 94032 Passau, Germany.

Journal of Geometric Analysis
|January 23, 2024
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Summary
This summary is machine-generated.

This study establishes new bounds for minimal spherical dispersion, showing its inverse is linear with ambient space dimension. These findings improve upon prior estimates for random point distributions on spheres.

Keywords:
DispersionExpected dispersionSpherical capSpherical dispersionVC-dimension

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Area of Science:

  • Mathematics
  • Geometric Measure Theory
  • Computational Geometry

Background:

  • Minimal spherical dispersion is a key concept in geometric analysis.
  • Previous estimates by Rote and Tichy (1995) provided foundational bounds.
  • Understanding dispersion is crucial for packing and covering problems.

Purpose of the Study:

  • To derive improved upper and lower bounds for minimal spherical dispersion.
  • To analyze the behavior of the inverse of minimal spherical dispersion with respect to dimension.
  • To establish bounds for expected dispersion of random points on a sphere.

Main Methods:

  • Mathematical analysis to derive theoretical bounds.
  • Comparison with existing estimates in geometric measure theory.
  • Probabilistic methods for analyzing random point distributions.

Main Results:

  • Established new upper and lower bounds for minimal spherical dispersion, refining previous work.
  • Demonstrated that the inverse of minimal spherical dispersion is linear in the ambient space dimension (d) for fixed epsilon.
  • Derived optimal bounds concerning epsilon for the expected dispersion of random points on the Euclidean unit sphere.

Conclusions:

  • The study provides a significant advancement in understanding minimal spherical dispersion.
  • The linearity of the inverse dispersion with dimension has implications for high-dimensional geometric problems.
  • The derived bounds offer precise estimates for random point configurations on spheres.