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Register a new userSlicing $ell_p$-balls reloaded: stability, planar sections in $ell_1$
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We show that the two-dimensional minimum-volume central section of the $n$-dimensional cross-polytope is attained by the regular $2n$-gon. We establish stability-type results for hyperplane sections of $ell_p$-balls in all the cases where the extremisers are known. Our methods are mainly probabilistic, exploring connections between negative moments of projections of random vectors uniformly distributed on convex bodies and volume of their sections.
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A recent result of Freeman, Odell, Sari, and Zheng states that whenever a separable Banach space not containing $ell_1$ has the property that all asymptotic models generated by weakly null sequences are equivalent to the unit vector basis of $c_0$ then the space is Asymptotic $c_0$. We show that if we replace $c_0$ with $ell_1$ then this result is no longer true. Moreover, a stronger result of B. Maurey - H. P. Rosenthal type is presented, namely, there exists a reflexive Banach space with an unconditional basis admitting $ell_1$ as a unique asymptotic model whereas any subsequence of the basis generates a non-Asymptotic $ell_1$ subspace.
Let $X$ be a sequence space and denote by $Z(X)$ the subset of $X$ formed by sequences having only a finite number of zero coordinates. We study algebraic properties of $Z(X)$ and show (among other results) that (for $p in [1,infty]$) $Z(ell_p)$ does not contain infinite dimensional closed subspaces. This solves an open question originally posed by R. M. Aron and V. I. Gurariy in 2003 on the linear structure of $Z(ell_infty)$. In addition to this, we also give a thorough analysis of the existing algebraic structures within the set $X setminus Z(X)$ and its algebraic genericity.
We study Banach spaces X with a strongly asymptotic l_p basis (any disjointly supported finite set of vectors far enough out with respect to the basis behaves like l_p) which are minimal (X embeds into every infinite dimensional subspace). In particular such spaces embed into l_p.
In this article we prove three fundamental types of limit theorems for the $q$-norm of random vectors chosen at random in an $ell_p^n$-ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guedon, Mendelson, and Naor. It includes the normalized volume and the cone probability measure as well as projections of these measures as special cases. Two new applications to random and non-random projections of $ell_p^n$-balls to lower-dimensional subspaces are discussed as well. The text is a continuation of [Kabluchko, Prochno, Thale: High-dimensional limit theorems for random vectors in $ell_p^n$-balls, Commun. Contemp. Math. (2019)].
James Tree Space ($mathcal{JT}$), introduced by R. James, is the first Banach space constructed having non-separable conjugate and not containing $ell^1$. James actually proved that every infinite dimensional subspace of $mathcal{JT}$ contains a Hilbert space, which implies the $ell^1$ non-embedding. In this expository article, we present a direct proof of the $ell^1$ non-embedding, using Rosenthals $ell^1$- Theorem and some measure theoretic arguments, namely Rieszs Representation Theorem.
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