Almost convex sets and best approximation.

We discuss notions of almost convexity of the following type: Let X be a Banach space and A be a nonempty subset. A is said to satisfy the condition Cp for a given p > 0 if for every x, y in A there is a z in A such that k(x+y)/2−zk "kx−ykp. We compare this property with related notions, continuing the work of several other authors. Moreover, we consider "(X) = sup{H(A, conv(A)): A unit ball of X, A satisfies C0(")}, where H(·, ·) is the Hausdorff distance. If X is B-convex (i.e. does not contain ln 1 ’s uniformly) then we show that for every e > 0 there is a constant c > 0 with "(X) +"c; here c depends only on and X, not ". If A is closed and ':A!R is lower bounded semicontinuous then the minimum problem {'(a): a 2 A} is said to be well posed if every minimizing sequence is convergent. In this case there is a unique minimum point for '(A). We give conditions such that the problem of best approximation of x by some element in A is well posed if x is close to A. (Here, of course, '(a) = kx−ak.) For example, let X = Lp, 2 p <1, and assume that A X satisfies Cp("). Then for every x in X with d(x,A) < (p−12−p"−1)1/(p−1) the best approximation problem for A is well posed.