Where do homogeneous polynomials on ℓ1n attain their norm?

Abstract

Using a 'reasonable' measure in Pð 2 c n 1 Þ; the space of 2-homogeneous polynomials on c n 1 ; we show the existence of a set of positive (and independent of n) measure of polynomials which do not attain their norm at the vertices of the unit ball of c n 1 : Next we prove that, when n grows, almost every polynomial attains its norm in a face of 'low' dimension.

A probabilistic approach to polynomial inequalities

“…We consider now the problem of where a polynomial P ∈ P ( k n 1 ) attains its norm. Pérez-García and Villanueva have proved in [6] that there is a set of positive measure (independent of the dimension n) of 2-homogeneous polynomials not attaining their norms on vertexes of the unit sphere.…”

“…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].…”