THE DIMENSION OF THE KERNEL IN PLANAR SETS WHICH ARE STAR-SHAPED VIA ALPHA-PATHS

Let S be a closed set in the plane and let alpha > 0. The following re sults hold for the alpha-kernel of S, denoted Ker(alpha)S. 1) When S i s bounded by a simple closed and locally connected curve, then Ker(alp ha)S has nonempty interior if and only if for some epsilon > 0, every 3 points of S see via alpha-paths in S some pair a, b with dist (a, b) greater than or equal to E. The number 3 is best possible. 2) When S is simply connected, then Ker(alpha)S = boolean AND{M : M a maximal al pha-convex subset of S}.