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From Browder [1] the following theorem is known: Let F be a closed subset of the Banach space E; then the set R of points x E aF, such that F n C = {x) for at least one convex C with nonempty interior, is dense in aF. A proof of this will be given by means of a theorem of Martin [4] on ordinary differential equations. Proofs of the just quoted result have been given by Browder [1], [2], Danes [3], and Phelps [5]. A completely different proof runs as follows: Assume the statement of Browder's theorem to be false. Then there exists a point p E aF and an open, convex neighbourhood U of p such that (1) RnaFnU=0. Now, for every function f: U -E the formula (2) lim inf IF, x + hf(x)l = O (x EaFn U) is valid (see below; IF, y I denotes the distance from F to the point y). Choose q E U \ F and define f(x)=_q-p (xE U). Then the unique solution of the initial value problem u(0) = p, u'(t) = f(u(t)) (O 0. Then Ceh. = {x + hf(x) + hesj0 < h < ho, Is 11 < 11 is a convex set with nonempty interior. By (1), x X R, and so there is some y E F nC Ceho, y x, i.e. y=x+hf(x)+hes, where 0<h < ho0, ls 1. Hence IF, x + hf(x)l < llx + hf(x) yll = heIIsII < he, yielding Received by the editors February 9, 1976. AMS (MOS) subject classifications (1970). Primary 46B99; Secondary 34G05. Copyright ? 1977, American Mathematical Society

New proof of a density theorem for the boundary of a closed set