BORIS DEKSTER - A version of the illumination problem

Let C be a convex n-dimensional body and d be a direction. A point is said to be illuminated by d if the line through p having the direction d contains a point which follows p. The body C is illuminated by a set of directions if each of its boundary points is illuminated by a . The illumination problem is to establish the minimum integer I_n such that, given C, one can choose a set of I_n directions illuminating C. This problem is open for . According to the Hadwiger conjecture, .

A convex n-dimensional body C is said to be exposable to if there is a linear transformation such that L(C) and each body isometric to L(C) is illuminated by D. Denote by E_n the minimum integer such that each C is exposable to a (fixed) set D of cardinality E_n. Obviously, since C is illuminated by L^-1(D) when C is exposable to D. (For n=2, I_n =4 and E_n =7). We establish here an upper bound for E_n. (This does not improve however either the Rogers and Zong estimate or the best known upper bound , due to Lassak.)

We introduce here also an affine invariant s, called sharpness, of a convex body. It ranges from (for smooth and some other convex bodies) to (for any n-simplex). Then we use it to specify our estimates. For instance, each convex n-body of sharpness can be illuminated by n+1 directions. This is an extension of a known Hadwiger's result saying that each smooth convex n-body is illuminated by n+1 directions.