Where do homogeneous polynomials on ℓ1n attain their norm? Original Research Article

Using a ‘reasonable’ measure in P(2ℓ1n), the space of 2-homogeneous polynomials on ℓ1n, 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 ℓ1n. Next we prove that, when n grows, almost every polynomial attains its norm in a face of ‘low’ dimension.