Convex Extension Preserving Lipschitz Constant

Let V;Rn be a non-empty bounded open and convex set and let f be a real function defined on the boundary of the set V. A necessary and sufficient condition is given for f to be extendable to a convex and lipschitzian function defined on the whole space Rn. The solution u to the degenerate case of the Monge]Amp´ere equation detw­ 2ur­ xi­ xj xs0, usf on the boundary of V, is lipschitzian if and only if f satisfies this condition