Let be the standard Euclidean norm on and let be a normed space. A subspace is strongly -Euclidean if there is a constant such that for every , and say that it is strongly -complemented if , where is the orthogonal projection from to and denotes the operator norm of with respect to the norm on . We give an example of a normed space of arbitrarily high dimension that is strongly 2-Euclidean but contains no 2-dimensional subspace that is both strongly -Euclidean and strongly -complemented, where 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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