Convex-transitive characterizations of Hilbert spaces

Abstract

In this paper we investigate real convex-transitive Banach spaces X, which admit a 1-dimensional bicontractive projection P on X. Various mild conditions regarding the weak topology and the geometry of the norm are provided, which guarantee that such an X is in fact isometrically a Hilbert space. The results obtained here can be regarded as partial answers to the wellknown Banach-Mazur rotation problem, as well as to a question posed by B. Randrianantoanina in 2002 about convex-transitive spaces.

Convex-transitivity and function spaces

“…See also [29] for similar results. (1) If Y is any Banach space such that dens(Y) < μ, then X contains an isometric copy of Y.…”

Locally uniformly rotund points in convex-transitive Banach spaces

“…B. Randrianantoanina [18] characterized Hilbert spaces as those almost transitive Banach spaces having a 1-complemented hyperplane. In Section 5, we slightly refine Randrianantoanina's theorem (see Corollary 5.3), and apply the main results in the present paper to rediscover and improve several "convex transitive" characterizations of Hilbert spaces obtained in [20] (see Theorem 5.5).…”

“…The version of Lemma 5.2 for real spaces appears as Theorem 2.5 of [18], and also as Proposition 1.3 of [20]. As pointed out in [20], this real version of Lemma 5.2 seems to be very old (see [2,Lemma 13.1 and (13.4 )]).…”

“…The version of Lemma 5.2 for real spaces appears as Theorem 2.5 of [18], and also as Proposition 1.3 of [20]. As pointed out in [20], this real version of Lemma 5.2 seems to be very old (see [2,Lemma 13.1 and (13.4 )]). Again in the real case, Corollary 5.3 becomes a slight refinement of the main result in [18] asserting that the Banach space X becomes a Hilbert space as soon as it is almost transitive and there exists a contractive linear projection from it onto some of its hyperplanes.…”

“…Following [20], given a vector space topology τ on X , we say that x is a τ -locally uniformly rotund (in short, τ -LUR) point of B X if for every sequence {x n } in B X such that lim n x + x n = 2 we have that τ -lim n x n = x. We note that LUR points of B X , as defined in the introduction, are precisely the τ -LUR points of B X for τ equal to the norm topology.…”