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Register a new userThe convex hull of a Banach-Saks set
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A subset $A$ of a Banach space is called Banach-Saks when every sequence in $A$ has a Ces{`a}ro convergent subsequence. Our interest here focusses on the following problem: is the convex hull of a Banach-Saks set again Banach-Saks? By means of a combinatorial argument, we show that in general the answer is negative. However, sufficient conditions are given in order to obtain a positive result.
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We prove a local version of Gowers Ramsey-type theorem [21], as well as loc
Let $K in R^d$ be a convex body, and assume that $L$ is a randomly rotated and shifted integer lattice. Let $K_L$ be the convex hull of the (random) points $K cap L$. The mean width $W(K_L)$ of $K_L$ is investigated. The asymptotic order of the mean width difference $W(l K)-W((l K)_L)$ is maximized by the order obtained by polytopes and minimized by the order for smooth convex sets as $l to infty$.
We generalize earlier results about connected components of idempotents in Banach algebras, due to B. SzH{o}kefalvi Nagy, Y. Kato, S. Maeda, Z. V. Kovarik, J. Zemanek, J. Esterle. Let $A$ be a unital complex Banach algebra, and $p(lambda) = prodlimits_{i = 1}^n (lambda - lambda_i)$ a polynomial over $Bbb C$, with all roots distinct. Let $E_p(A) := {a in A mid p(a) = 0}$. Then all connected components of $E_p(A)$ are pathwise connected (locally pathwise connected) via each of the following three types of paths: 1)~similarity via a finite product of exponential functions (via an exponential function); 2)~a polynomial path (a cubic polynomial path); 3)~a polygonal path (a polygonal path consisting of $n$ segments). If $A$ is a $C^*$-algebra, $lambda_i in Bbb R$, let $S_p(A):= {ain A mid a = a^*$, $p(a) = 0}$. Then all connected components of $S_p(A)$ are pathwise connected (locally pathwise connected), via a path of the form $e^{-ic_mt}dots e^{-ic_1t} ae^{ic_1t}dots e^{ic_mt}$, where $c_i = c_i^*$, and $t in [0, 1]$ (of the form $e^{-ict} ae^{ict}$, where $c = c^*$, and $t in [0,1]$). For (self-adjoint) idempotents we have by these old papers that the distance of different connected components of them is at least~$1$. For $E_p(A)$, $S_p(A)$ we pose the problem if the distance of different connected components is at least $min bigl{|lambda_i - lambda_j| mid 1 leq i,j leq n, i eq jbigr}$. For the case of $S_p(A)$, we give a positive lower bound for these distances, that depends on $lambda_1, dots, lambda_n$. We show that several local and global lifting theorems for analytic families of idempotents, along analytic families of surjective Banach algebra homomorphisms, from our recent paper with B. Aupetit and M. Mbekhta, have analogues for elements of $E_p(A)$ and $S_p(A)$.
Number Decision Diagrams (NDD) provide a natural finite symbolic representation for regular set of integer vectors encoded as strings of digit vectors (least or most significant digit first). The convex hull of the set of vectors represented by a NDD is proved to be an effectively computable convex polyhedron.
The purpose of this article is to present the construction and basic properties of the general Bochner integral. The approach presented here is based on the ideas from the book The Bochner Integral by J. Mikusinski where the integral is presented for functions defined on $mathbb{R}^N$. In this article we present a more general and simplified construction of the Bochner integral on abstract measure spaces. An extension of the construction to functions with values in a locally convex space is also considered.
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