Where do homogeneous polynomials on ln1 attain their norm?
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2004
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Elsevier
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Pérez Garcı́a, D. & Villanueva Díez, I. «Where Do Homogeneous Polynomials on ℓ1n Attain Their Norm?» Journal of Approximation Theory, vol. 127, n.o 1, marzo de 2004, pp. 124-33. DOI.org (Crossref), https://doi.org/10.1016/j.jat.2004.01.001.
Abstract
Using a ‘reasonable’ measure in , 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.