Related papers: Covering $B_X$ by finitely many convex sets

We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach spaces. We prove that the norm of an arbitrary Banach space is octahedral if, and only if, every convex combination of…

For Banach spaces of analytic functions on the disc for which the polynomials are dense and their pointt evaluations continuous, we prove the following: If they contain a function such that the limit superior of its modulus is infinite…

We study linear and algebraic structures in sets of bounded holomorphic functions on the ball which have large cluster sets at every possible point (i.e., every point on the sphere in several complex variables and every point of the closed…

For each positive integer $m$ and each real finite dimensional Banach space $X$, we set $\beta(X,m)$ to be the infimum of $\delta\in (0,1]$ such that each set $A\subset X$ having diameter $1$ can be represented as the union of $m$ subsets…

The following strengthening of the Elton-Odell theorem on the existence of a $(1+\epsilon)-$separated sequences in the unit sphere $S_X$ of an infinite dimensional Banach space $X$ is proved: There exists an infinite subset $S\subseteq S_X$…

We study the class of compact convex subsets of a topological vector space which admits a strictly convex and lower semicontinuous function. We prove that such a compact set is embeddable in a strictly convex dual Banach space endowed with…

We derive an upper bound on the size of a ball such that the image of the ball under quadratic map is strongly convex and smooth. Our result is the best possible improvement of the analogous result by Polyak in the case of quadratic map. We…

We fully characterize those von Neumann algebras having the ball-covering property. We also study the ball-covering property of noncommutative symmetric spaces. In particular, we provide a number of new examples of non-separable…

Let $E$ be a Banach space and $\X$ be the closed unit ball of the dual space $E^*$. For a compact set $K$ in $E$, we prove that $K$ is polynomially convex in $E$ if and only if there exist a unital commutative Banach algebra $A$ and a…

In this paper, we improve the lower estimate of multidimensional Bohr radius for unit ball of $\ell^n_q$-spaces ($1\leq q\leq \infty$) for bounded holomorphic functions with values in finite dimensional complex Banach spaces. The new…

We extend a classical result by Rankin. We consider the following question: given $n$ vectors $v_i$ in the ball of radius $R$ of an infinite dimensional Banach space ${\cal B}$ with $d(v_i,v_j)\geq 1$, can we bound the number $n$?

Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric…

We show that a pseudoconvex open subset of a Banach space with an unconditional basis is biholomorphic to a closed direct submanifold of a Banach space with an unconditional basis.

It is well known that finite-dimensional polyhedral convex sets can be generated by finitely many points and finitely many directions. Representation formulas in this spirit are obtained for convex polyhedra and generalized convex polyhedra…

We consider the problem of wrapping three-dimensional solid bodies with a given planar sheet of paper, where the paper may be folded or wrinkled but not stretched or torn. We propose a conjecture characterising the maximumvolume solid…

If (M^n, g) is a complete Riemannian manifold with filling radius at least R, then we prove that it contains a ball of radius R and volume at least c(n)R^n. If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M with…

A reflexive Banach space $X$ with a basis $(e_i)$ is constructed having the property that every monotone basis is block finitely representable in each block basis of $X$.

We prove that if $B\subseteq A$ is an extension of finite dimensional algebras such that the projective dimension of $A/B$ as a $B$-bimodule is finite, if $A$ has finite finitistic dimension, then so does $B$. We exhibit examples…

We introduce and study the notion of overcomplete set in a Banach space, that subsumes and extends the classical concept of overcomplete sequence in a (separable) Banach space. We give existence and non-existence results of overcomplete…

A closed 4-manifold (or, more generally, a finite $PD_4$-space) has a finitely dominated infinite regular covering space if and only if either its universal covering space is finitely dominated or it is finitely covered by the mapping torus…