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A fundamental result in the theory of convex functions [Bo, p.60] states that any locally bounded above convex function on an open convex set is continuous. Already the closed interval [0,1] shows that this conclusion is not necessarily valid if the convex set is not open. However, it is wellknown [Ro, p. 84] that the interval [0,1], and more generally any closed convex polytope P, has the property that any locally bounded above convex function on P is upper semi-continuous. We came across convex sets with this property in the following way: Let K be any non-empty convex set in a locally convex topological vector space, let e be a point outside the linear subspace generated by the closure of K, let K* be the convex envelope of K and e. Choose x 0 E K and let II(·, x0 ) be the affine projection from K* onto J( which sends e to xo. We ask: When is II(·,xo) an open map whenever x0 E K? Our answer is that this is true if and only if K has the property that any locally bounded above convex function on K is upper semi-continuous. A convex set with this property will be called an excellent convex set. Note, that by the preceding, every open convex set and every closed convex polytope is an excellent set. In fact, we shall prove in section 1 that the closed convex polytopes are the only compact convex sets that are excellent. There is a geometric characterization of this property: A convex set K is excellent if and only if for any x0 E J( and any homothetie h>.(·,x0 ) with center x0 and factor A. E< 0,1 >,the image h>.(K,x0 ) of J( is a neighborhood of x0 in K.(Theorem 1.13). We discovered these two charactrizations of excellent convex sets with the help of a certain function A(·, xo) defined on K by

Upper Semi-Continuity of Convex Functions and Openness of Affine Maps.