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Covering functionals of cones and double cones.

Senlin Wu1, Ke Xu1

  • 1Department of Applied Mathematics, Harbin University of Science and Technology, Harbin, China.

Journal of Inequalities and Applications
|August 24, 2018
PubMed
Summary

Researchers estimated covering functionals for convex bodies, specifically cones and double cones. These estimations are the best possible for certain parameters, advancing quantitative geometry research.

Keywords:
Banach–Mazur distanceCompact convex conesConvex bodyCoveringHadwiger’s covering conjecture

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Area of Science:

  • Convex Geometry
  • Geometric Functional Analysis

Background:

  • The study addresses Chuanming Zong's quantitative program for Hadwiger's covering conjecture.
  • Covering functionals are crucial for estimating how convex bodies can be covered by their scaled translates.

Purpose of the Study:

  • To estimate the covering functional () for convex bodies, particularly cones and double cones.
  • To provide best possible estimations for specific pairs of parameters (m, K).

Main Methods:

  • The study involves geometric analysis and the calculation of covering functionals.
  • Specific geometric properties of cones and double cones are utilized.

Main Results:

  • New estimations for the covering functionals of cones and double cones are presented.
  • These estimations are shown to be optimal for certain configurations of m and K.

Conclusions:

  • The findings contribute to the quantitative understanding of covering properties of convex bodies.
  • The presented estimations advance the research program related to Hadwiger's covering conjecture.