A counterexample to a strengthening of a question of V. D. Milman
Annales Henri Lebesgue, Volume 6 (2023), pp. 427-448.


Keywords normed space, almost Euclidean, well complemented


Let |·| be the standard Euclidean norm on n and let X=( n ,·) be a normed space. A subspace YX is strongly α-Euclidean if there is a constant t such that t|y|yαt|y| for every yY, and say that it is strongly α-complemented if P Y α, where P Y is the orthogonal projection from X to Y and P Y denotes the operator norm of P Y with respect to the norm on X. We give an example of a normed space X of arbitrarily high dimension that is strongly 2-Euclidean but contains no 2-dimensional subspace that is both strongly (1+ϵ)-Euclidean and strongly (1+ϵ)-complemented, where ϵ>0 is an absolute constant. This property is closely related to an old question of Vitali Milman. The example is probabilistic in nature.


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