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.