On the Regularity of the Convexiflcation Operator on a Compact Set

Let E denote a real vector space and X be a convex subset of E. Denote by B(X) the set of bounded real valued functions on X. The convexification operator on X, coX (·) , is from B(X) to itself. It associates to a function f the greatest convex function smaller than f . Explicitly, for f in B(X) and x in X, coX (f) (x) := sup {g(x) : g ∈ S(X), g (·) ≤ f (·)} . where S(X) denotes the set of bounded real valued convex functions onX and g (·) ≤ f (·) means that g(y) ≤ f(y) for any y in X.