We consider the problem of existence of polynomials with small norm. This range of problems has been extensively studied by many authors in the case of the unit circle (or a compact Abelian group), i.e. when the characters are bounded. In general, on compact homogeneous Riemannian manifolds, the eigenfunctions of the Laplace-Beltrami operator are not uniformly bounded. This creates difficulties of a fundamental nature in applications of known methods and results. The method we develop is based on a geometric inequality between norms induced by two convex bodies in Euclidean space.