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For subsets of $mathbb{R}^+$ we consider the local right upper porosity and the local right lower porosity as elements of a cluster set of all porosity numbers. The use of a scaling function $mu:mathbb{N} to mathbb{R}^+$ provides an extension of the concept of porosity numbers on subsets of $mathbb{N}$. The main results describe interconnections between porosity numbers of a set, features of the scaling functions and the geometry of so-called pretangent spaces to this set.
Let $X$ be a geodesic metric space with $H_1(X)$ uniformly generated. If $X$ has asymptotic dimension one then $X$ is quasi-isometric to an unbounded tree. As a corollary, we show that the asymptotic dimension of the curve graph of a compact, oriente
We prove that on an essentially non-branching $mathrm{MCP}(K,N)$ space, if a geodesic ball has a volume lower bound and satisfies some additional geometric conditions, then in a smaller geodesic ball (in a quantified sense) we have an estimate on the isoperimetric constants.
We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) exotic isometric flow. This contrast
In this article, we show that the Goldman-Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls. On the non-Archimedean side, we deduce that most classical Bruhat-Tits buildings may be endowed with a
Let $(X, d)$ be a compact metric space and let $mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I colon mathcal{M}(X) to R$ by $I(mu) = int_X int_X d(x,y) dmu(x) dmu(y)$, and set $M(X) = sup I(mu)$, where $mu$ rang