  By the celebrated Weierstrass Theorem the set of algebraic polynomials is
dense in the space of continuous functions on a compact set in R^d. In this
paper we study the following question: does the density hold if we approximate
only by homogeneous polynomials? Since the set of homogeneous polynomials is
nonlinear this leads to a nontrivial problem. It is easy to see that: 1)
density may hold only on star-like origin-symmetric surfaces; 2) at least 2
homogeneous polynomials are needed for approximation. The most interesting
special case of a star-like surface is a convex surface. It has been
conjectured by the second author that functions continuous on origin-symmetric
convex surfaces in R^d can be approximated by a pair of homogeneous
polynomials. This conjecture is not resolved yet but we make substantial
progress towards its positive settlement. In particular, it is shown in the
present paper that the above conjecture holds for 1) d=2, 2) convex surfaces in
R^d with C^(1+epsilon) boundary.

Classical Analysis and ODEs