Consider a graph optimization on a graph with n vertices and m edges that can be written as an LP (like say bipartite matching). By the duality with vertex cover, we know that there's a sparse dual solution consisting of the elements of the cover. It's sparse in the sense that its size is linear in n, rather than in m.
Consider the problem of finding the minimum enclosing ball of a set of $n$ points in $d$ dimensions. This is an LP-type problem with (again) a sparse dual solution: a set of $d+1$ points that define the optimal ball. Here, "sparse" means that the size depends on the (smaller) parameter $d$ rather than $n$
In general, suppose I have a problem that can be expressed via an LP with $n$ variables and $m$ constraints, where $m \gg n$. Are there classes of such problems for which any optimal solution to the problem can be expressed using the intersection of only $O(n)$ constraints (or dually with only $O(n)$ nonzero dual variables ?