On Nonconvexity of Graphs of Polynomials of Several Real Variables

We consider transversal (orthogonal) perturbations of finite-dimensional convex sets and estimate the degree of nonconvexity of resulting sets, i.e. we estimate the nonconvexity of graphs of continuous functions. We prove that a suitable estimate of nonconvexity of graphs over all lines induces a nice estimate of the nonconvexity of graphs of the entire function. Here, the term nice means that in the well-known Michael selection theorem it is possible to replace convex sets of a multivalued mapping by such nonconvex sets. As a corollary, we obtain positive results for polynomials of degree two under some restrictions on coefficients. Our previous results concerned the polynomials of degree one and Lipschitz functions. We show that for a family of polynomials of degree three such estimate of convexity in general does not exist. Moreover, for degree 9 we show that the nonconvexity of the unique polynomial P(x,y)=x⁹+x³y realizes the worst possible case.