On Max- K -Sums

The max- k -sum of a set of real scalars is the maximum sum of a subset of size k , or alternatively the sum of the k largest elements. We study two extensions: first, we show how to obtain smooth approximations to functions that are pointwise max- k -sums of smooth functions. Second, we discuss how the max- k -sum can be defined on vectors in a finite-dimensional real vector space ordered by a closed convex cone.