Boris Dekster - Each convex body in E3 symmetric about a plane can be illuminated by 8 directions

Next: Paul Gauthier - The Up: Contributed Papers / Communications Previous: Nataliya Bantsur - Existence

Boris Dekster - Each convex body in E3 symmetric about a plane can be illuminated by 8 directions

BORIS DEKSTER, Mount Allison University, Sackville, New Brunswick E4L 1E6

Each convex body in E3 symmetric about a plane can be illuminated by 8 directions

Let C be a convex body in E^d, $d \geq 2$ . Let x be a point on $\partial C$ and v be a direction (non-zero vector). Consider the axis l having the direction v and passing through x. The direction v is said to illuminate x if l contains a point $y \in {\rm int}\,C$ which succeeds x. If each point of a part of $\partial C$ is illuminated by at least one of a few directions, the body C is said to be illuminated by these directions. We prove the following.

Theorem convex body in E³ symmetric about a plane can be illuminated by 8 directions.

For polyhedral bodies, this result was established by K. Bezdek in 1991. Our method however is quite different. Both results are partial proofs of the Hadwiger Conjecture according to which each convex body in E^d can be illuminated by 2^d directions.

Next: Paul Gauthier - The Up: Contributed Papers / Communications Previous: Nataliya Bantsur - Existence