Error Bounds for Convex Polynomials

The purpose of this paper is to investigate error bounds for convex polynomials. We prove that for a convex polynomial $f$ in $n$ variables which is not everywhere positive and which is not constant on any affine subspace, either $f$ is a sum of a convex polynomial in fewer variables and a linear form with negative coefficients or the negativity set of $f$ is compact. As an application, we deduce various types of error bounds for unconstrained and polyhedral-constrained convex polynomials.

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