Motivated by work in number theory, in the 1970s Waldschmidt defined an asymptotic measure of the least degree of a polynomial in n variables with given order of vanishing on a finite set of points in projective space. In the case of generic points in P2, determining the value of Waldschmidt's constant is equivalent to an open conjecture of Nagata. Recent work has related Waldschmidt's constant to an ideal containment problem: which symbolic powers of the ideal of the points are contained in a given power of the ideal? In joint work with E. Guardo and A. Van Tuyl, we generalize this work from points to lines in projective space and we formulate a version of Nagata's conjecture in this context. Additional joint work with M. Dumnicki, T. Szemberg and H. Tutaj-Gasinska extends this to r-planes in projective N-space.