On the typical structure of compact sets

Let E be a strictly convex separable Banach space of dimension at least 2. A compact set K subset of E has hispid structure if the nearest point mappi ng p(K) : E --> 2(K) is not single valued on a dense subset of E. It is pro ved that if K is a union of disjoint compact sets K-1, K-2 and K has hispid structure, then K-1 and K-2 have hispid structure. Furthermore. it is show n that for typical tin the sense of Baire category) compact set K and arbit rary x is an element of E the set of all r > 0 such that the intersection K boolean AND B(x, r) not equal 0 and has no hispid structure is of Jordan m easure zero.