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BORIS DEKSTER - A version of the illumination problem
BORIS DEKSTER, Department of Mathematics and Computer Science, Mount Allison University, Sackville, New Brunswick E4L 1E6, Canada | |
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 In such that, given C, one can choose a set of In 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 En the minimum integer such that each C is exposable to a (fixed) set D of cardinality En. Obviously, since C is illuminated by L-1(D) when C is exposable to D. (For n=2, In =4 and En =7). We establish here an upper bound for En. (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.
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