Estimating L8 Norms by L2k Norms for Functions on Orbits

Let G be a compact group acting in a real vector space V. We obtain a number of inequalities relating the L 1 norm of a matrix element of the representation of G with its L 2k norm for a positive integer k. As an application, we obtain approximation algorithms to nd the maximum absolute value of a given multivariate polynomial over the unit sphere (in which case G is the orthogonal group) and for the assignment problem of degree d, a hard problem of combinatorial optimization generalizing the quadratic assignment problem (in which case G is the symmetric group).