Where do homogeneous polynomials attain their norm?

Abstract

Given a fixed subset A of the unit sphere of a real finite-dimensional Banach space, how probable is it for a norm-one homogeneous polynomial to attain its norm on A? We study the linear case in Section 1, and in Section 2 consider the case of k-homogeneous polynomials on n ∞ .

A probabilistic approach to polynomial inequalities

“…One may argue -by considering gradients of a polynomial at different points and by comparing the 2 -norms of vertexes vis-a-vis other points of the unit sphere of n ∞ -that homogeneous polynomials over n ∞ are more likely to attain their norms at vertexes than at any other point, and that such a likelihood will grow as the dimension n tends to infinity. This has been partially addressed in [3] and in [6].…”