Normal and starlike tilings in separable Banach spaces

Normal tilings of a Banach space and its ball

We show some new results about tilings in Banach spaces. A tiling of a Banach space $X$ is a covering by closed sets with non-empty interior so that they have pairwise disjoint interiors. If moreover the tiles have inner radii uniformly bounded from below, and outer radii uniformly bounded from above, we say that the tiling is normal. In 2010 Preiss constructed a convex normal tiling of the separable Hilbert space. For Banach spaces with Schauder basis we will show that Preiss' result is still true with starshaped tiles instead of convex ones. Also, whenever $X$ is uniformly convex we give pre…

Convexity properties of quasihyperbolic balls on Banach spaces

We study convexity and starlikeness of quasihyperbolic and distance ratio metric balls on Banach spaces. In particular, problems related to these metrics on convex domains, and on punctured Banach spaces, are considered.

Rolle's theorem is either false or trivial in infinite-dimensional Banach spaces

We prove the following new characterization of $C^p$ (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space $X$ has a $C^p$ smooth (Lipschitz) bump function if and only if it has another $C^p$ smooth (Lipschitz) bump function $f$ such that $f'(x)\neq 0$ for every point $x$ in the interior of the support of $f$ (that is, $f$ does not satisfy Rolle's theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. As a by-product of the proof of this result we also obtain other useful characterizations of $C^p$ smoothness related to the existence o…

在ℓ₁中的球形镶嵌与覆盖

Tilings and coverings by balls in $\ell_1$

本文研究了ℓ₁空间中球形镶嵌的存在性，证明了在可数维空间中无法实现球形镶嵌，并探讨了星-有限覆盖的性质。

多面体赋范空间：结构定理与局部有限铺砌

Polyhedral normed spaces: the structural theorem and locally finite tilings

本文证明具有($\Delta$)性质的多面体赋范空间的单位球面被代数真面覆盖，并推广到(VI)-多面体情形，进而刻画了局部有限铺砌的存在性，推广了Fonf的结果。

希尔伯特空间的格铺砌

Lattice tilings of Hilbert spaces

本文通过构造 $\ell_2(\Gamma)$ 中的有界对称凸体，首次给出无穷维自反巴拿赫空间允许半径为1的球格铺砌的例子，并进一步研究了格铺砌的相交性质及赋范空间离散子群的结构。