On Nonconvexity of Graphs of Polynomials of Several Real Variables

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Abstract

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.

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