5

This question is a follow-on question from:
Covering a unit ball with balls half the radius

The questions are these: Given an arbitrary dimension d, and a unit n-ball in d-dimensional Euclidean space...

  • Is the fewest number of n-balls of equal radius (r < 1) that can cover the unit n-ball always d + 1?

YES, the problem of covering using d+1 n-balls can be viewed as a covering using the simplest regular n-dimensional polytope, the regular n-simplex. The covering n-ball centers are placed on the 0-faces of the simplex inscribed into the n-ball inscribed into the n-simplex inscribed into the unit n-ball. Proven by K. Böröczky, see pg 189: http://books.google.com/books?id=sS7FKuuM33oC

  • Is the smallest radius of these n-balls sqrt(3) / 2 regardless of dimension?

YES FOR INTEGER DIMENSIONS >= 2, the radii of the d+1 covering n-balls approach 1 from sqrt(3) / 2 for d in the limit. The covering n-balls are centered at a distance of 1/d from the origin, having a tetrahedral angle of arccos(-1/d). Proven in private communication, contact for details (trivial via above).

  • Is the second fewest number of n-balls of equal size (r < 1) that can cover the unit n-ball always 2d?

YES, the problem of covering using 2d n-balls can be viewed as a covering using the second simplest regular n-dimensional polytope, the regular cross-polytope. The covering n-ball centers are placed on the 0-faces of the cross-polytope inscribed into the n-ball inscribed into the cross-polytope inscribed into the unit n-ball. Minimum second optimal covering proven by Barton and Bird in private communication (non-trivial).

  • Is the smallest radius of these n-balls sqrt(2) / 2 regardless of dimension?

YES FOR INTEGER DIMENSIONS >= 2, the radii of the 2d covering n-balls approach 1 from sqrt(2) / 2 for d in the limit. The n-balls are centered at a distance of 1/d from the origin, having a tetrahedral angle of arccos((2 - d) / d). Proven in private communication (trivial via above).

An interesting additional question is:
- Is there a derivable dimension-dependent formula for the number of 0.5 radius n-balls required for a complete covering? A number of data points exist (e.g. 7 in 2D, 21 in 3D)

OPEN

Community
  • 1
Rob Bird
  • 151
  • a) Think of the regular simplex. c) Think of how the center is lost in a). b,d) Think of the volume. e) Very unlikely, but who knows... – fedja May 31 '13 at 23:52
  • Can you explain what you mean by "think of the regular simplex"? – Rob Bird Jun 01 '13 at 02:21
  • Vague, but excellent advice! The circumscribing simplex proves to be the tool for solving parts 1 and 2. – Rob Bird Jun 27 '13 at 11:46

0 Answers0