STRICT VERIFICATION OF APPROXIMATE MIDCONVEXITY ON NON-CONVEX SETS

Let V be a subset of an Abelian group G and let omega : V x V -> [0, infinity] be given. We say that a function f : V -> R is omega(.,.)-midconvex if f (x) <= f (x - delta)+ f (x + delta)/2 + omega(x - delta, x + delta) for x is an element of V, delta is an element of G such that x - delta, x + delta is an element of V. Our aim is to provide a computer assisted method to estimate sup{f is an element of V -> R: f is an element of B(V;W), f is omega(.,.)-midconvex}, where B(V;W) denotes the set of real-valued, bounded from above functions on V which are zero on W (W subset of V). We present an algorithm which for given epsilon > 0 enables us, under reasonable assumptions, to find the above supremum with accuracy epsilon. We test our results for V = {0, 1/N, ... , N-1/N, 1} and W = {0,1}, where N is an element of N is fixed.