WHERE DO HOMOGENEOUS POLYNOMIALS ON ℓ_{1}^{n} ATTAIN THEIR NORM?

DAVID PÉREZ-GARCÍA AND IGNACIO VILLANUEVA

Abstract. In the “Open Problems” session of the conference ‘Function Theory on Infinite Dimensional Spaces VII’, held in Madrid in 2001, Professor I. Zalduendo asked the question of ‘how many’ homogeneous polynomials will attain their norm at the vertices of the unit ball of ℓ_{∞}^{n} when n tends to infinity. He conjectured that ‘almost everyone’. In this direction, he and D. Carando published recently a paper giving qualitative general results. As they say in the introduction, the question is to study how likely it is for a polynomial P : E ⟶ ℝ to attain its norm at a given subset A of the unit ball B_{E}.

In our talk we present quantitative results referring to 2-homogeneous poynomials on ℓ_{1}^{n}, as an example of the results that can be expected in more general cases: Using a ‘reasonable’ measure in P(^{2}ℓ_{1}^{n}), the space of 2-homogeneous polynomials on ℓ_{1}^{n}, 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 ℓ_{1}^{n}. Next we prove that, when n grows, almost every polynomial attains its norm in a face of ‘low’ dimension.