Best approximation from the intersection of a closed convex set and a polyhedron in Hilbert space, weak slater conditions, and the strong conical hull intersection property

Let X be a (real) Hilbert space, C be a closed convex subset, and H-i = {x is an element of X \ [x, h(i)] less than or equal to b(i)} (i = 1, 2,..., m ) be a finite collection of half-spaces. Under the assumption that K := C b oolean AND (boolean AND(1)(m) H-i) is not empty, the problem of characteriz ing the best approximation from K to any x is an element of X is considered . The "strong conical hull intersection property" (strong CHIP), which was introduced by us in 1997, is shown to be both necessary and sufficient for the following "perturbation property" to hold: for each x 2 X, an element x (0) is an element of K satisfies x(0) = PK (x) if and only if x(0) = P-C(x - Sigma(1)(m) lambda(i)h(i)) for some scalars lambda(i) greater than or equ al to 0 with lambda(i)[[x(0), b(i)] - b(i)] = 0 for each i. Here P-D(z) den otes the unique best approximation from D to z. In other words, determining the best approximation from the set K to any point is equivalent to the (g enerally easier) problem of determining the best approximation from the set C to a perturbation of that point. Moreover, even when the strong CHIP doe s not hold, the perturbation property still holds, except now C must be rep laced by a certain convex extremal subset of C. We also show that the stron g CHIP is weaker than any of the weak Slater conditions that one can natura lly impose on the sets in question. These results generalize the main resul ts of our 1997 paper [J. Approx. Theory, 90, pp. 385-444] and hence those o f several other papers as well.