Publications

Let D be a subset of R^n with positive finite Lebesgue measure and f on D an arbitrary real summable function. Then the function F of a, defined by the integral of f - a over all x of D with f(x) larger as or equal to a, is continuous, non-negative, non-increasing, convex, and has almost everywhere the measure of the level set as derivative F'(a). Further on, it holds that the essential supremum of f is given by the supremum of all a with F(a)>0. These properties can be used for computing the essential supremum of f. As example, two algorithms are stated. If the function f is dense, or lower semicontinuous, or if -f is robust, then supremum and the essential supremum of f coincide. In this case, the algorithms mentioned can be applied for determining the supremum of f, i.e., also the global maximum of f if it exists.