Functionally closed sets and functionally convex sets in real Banach spaces

In 1965, L.P. Vlasov dened an approximately convex subset M of a linear normed space X, by denoting the multivalued mapping which puts into correspondence with each point x 2 X, the set Tx of all points y 2 M which satisfy the condition d(x; y) = d(x;M). Then the set M is called approximately convex if, for x 2 X the set Tx is nonempty and convex. He proved that, in Banach spaces which are uniformly smooth in each direction, each approximately compact and approximately convex set is convex .