A Fixed-point Theorem for Inward and Outward Maps

The Schauder-Tychonoff theorem states that a continuous function from a compact convex subset of a locally convex topological vector space into itself must have a fixed point ([1, Chapter V, 10.5], or [2]). Using this theorem, we obtain here a stronger result, stating that a map from such a set into the surrounding vector space has a fixed point if the directions in which the points are moved satisfy a certain "inwardness" condition. It follows immediately that a symmetrical "outwardness" condition also implies the existence of a fixed point. We find also that under the latter condition the image of the map necessarily includes the original set!