arXiv subjects
Publications and source records attributed to G. Vassiliadis.
We study some touching properties of the three-dimensional Petty space $X=(\ell_2^2 \oplus \mathbb{R})_1$. In particular we give an estimation of its Hadwiger number and also show that its equilateral subsets $A$ of maximum cardinality (i.e. $|A|=e(X)$) do not have a center.
In this paper we apply ideas from the theory of Uniform Distribution of sequences to Functional Analysis and then drawing inspiration from the consequent results, we study concepts and results in Uniform Distribution itself. So let $E$ be a Banach space. Then we prove:\\ (a) If $F$ is a bounded subset of $E$ and $x \in \overline{\co}(F)$ (= the closed convex hull of $F$), then there is a sequence $(x_n) \subseteq F$ which is Cesàro summable to $x$.\\ (b) If $E$ is separable, $F \subseteq E^*$ bounded and $f \in \overline{\co}^{w^*}(F)$, then there is a sequence $(f_n) \subseteq F$ whose sequence of arithmetic means $\frac{f_1+\dots+f_N}{N}$, $N \ge 1$ weak$^*$-converges to $f$. By the aid of the Krein-Milman theorem, both (a) and (b) have interesting implications for closed, convex and bounded subsets $Ω$ of $E$ such that $Ω=\overline{\co}(\ex Ω)$ and for weak$^*$ compact and convex subsets of $E^*$. Of particular interest is the case when $Ω=B_{C(K)^*}$, where $K$ is a compact metric space. By further expanding the previous ideas and results, we are able to generalize a classical theorem of Uniform Distribution which is valid for increasing functions $φ:I=[0,1] \rightarrow \mathbb{R}$ with $φ(0)=0$ and $φ(1)=1$, for functions $φ$ of bounded variation on $I$ with $φ(0)=0$ and total variation $V_0^1 φ=1$.
Let $X$ be a finite-dimensional Banach space; we introduce and investigate a natural generalization of the concepts of Hadwiger number $H(X)$ and strict Hadwiger number $H'(X)$. More precisely, we define the antipodal Hadwiger number $H_α(X)$ as the largest cardinality of a subset $S \subseteq S_X$, such that $\forall x \neq y \in S \,\,\, \exists f \in B_{X^*}$ with \[1 \le f(x)-f(y) \,\,\, \textrm{and} \,\,\, f(y) \le f(z) \le f(x) \,\,\, \textrm{for} \,\,\, z \in S.\] The strict antipodal Hadwiger number $H'_α(X)$ is defined analogously. We prove that $H'_α(X)=4$ for every Minkowski plane and estimate (or in some cases compute) the numbers $H_α(X)$ and $H'_α(X)$, where $X=\ell_p^n, 1 < p \le +\infty$ and $n \ge 2$. We also show that the number $H'_α(X)$ grows exponentially in $\dim X$.
Let $(M,d)$ be a bounded countable metric space and $c>0$ a constant, such that $d(x,y)+d(y,z)-d(x,z) \ge c$, for any pairwise distinct points $x,y,z$ of $M$. For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of $\ell_\infty$.
We show that for "most" compact non metrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact non metrizable spaces K such that the minimum cardinality of a maximal equilateral set in C(K) is countable.
We show that every Banach space $X$ containing an isomorphic copy of $c_0$ has an infinite equilateral set and also that if $X$ has a bounded biorthogonal system of size $α$ then it can be renormed so as to admit an equilateral set of equal size. If $K$ is any compact non metrizable space, then under a certain combinatorial condition on $K$ the Banach space $C(K)$ has an uncountable equilateral set.