On Chebyshev functions and Klee functions

Associated to a lower semicontinuous function, one can define its proximal mapping and farthest mapping. The function is called Chebyshev (Klee) if its proximal mapping (farthest mapping) is single-valued everywhere. We show that the function f is 1 / λ -hypoconvex if its proximal mapping P λ f is single-valued. When the function f is bounded below, and P λ f is single-valued for every λ > 0 , the function must be convex. Similarly, we show that the function f is 1 / μ -strongly convex if the farthest mapping Q μ f is single-valued. When the function is the indicator function of a set, this recovers the well-known Chebyshev problem and Klee problem in R n . We also give an example illustrating that a continuous proximal mapping (farthest mapping) needs not be locally Lipschitz, which answers one open question by Hare and Poliquin.