In this paper,we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called RCD*(K, N) spaces) with non-empty one dimensional regular sets. In particular, we prove that the class of Ricci limit spaces with Ric ≥ K and Hausdorff dimension N and the class of RCD*(K, N) spaces coincide for N < 2 (They can be either complete intervals or circles). We will also prove a Bishop-Gromov type inequality (that is ,roughly speaking, a converse...

For every metric space X we introduce two cardinal characteristics $co{v}^{♭}\left(X\right)$ and $co{v}^{♯}\left(X\right)$ describing the capacity of balls in X. We prove that these cardinal characteristics are invariant under coarse equivalence, and that two ultrametric spaces X,Y are coarsely equivalent if $co{v}^{♭}\left(X\right)=co{v}^{♯}\left(X\right)=co{v}^{♭}\left(Y\right)=co{v}^{♯}\left(Y\right)$. This implies that an ultrametric space X is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if $co{v}^{♭}\left(X\right)=co{v}^{♯}\left(X\right)$. Moreover, two isometrically homogeneous ultrametric spaces X,Y are coarsely equivalent if and only if $co{v}^{♯}\left(X\right)=co{v}^{♯}\left(Y\right)$...

This paper discusses properties of the Higson corona by means of a quotient on coarse ultrafilters on a proper metric space. We use this description to show that the corona functor is faithful and reflects isomorphisms.

The definition of n-width of a bounded subset A in a normed linear space X is based on the existence of n-dimensional subspaces. Although the concept of an n-dimensional subspace is not available for metric trees, in this paper, using the properties of convex and compact subsets, we present a notion of n-widths for a metric tree, called Tn-widths. Later we discuss properties of Tn-widths, and show that the compact width is attained. A relationship between the compact widths and Tn-widths is also...

Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of fiite metric spaces into L1, there is a similar, yet undeveloped, theory for embedding finite diversities into the diversity analogue of L1 spaces. In the metric case, it iswell known that an n-point metric space can be embedded into L1 withO(log n) distortion. For diversities, the optimal distortion is unknown....

The quotient singularities of dimensions two and three obtained from polyhedral groups and the corresponding binary polyhedral groups admit natural resolutions of singularities as Hilbert schemes of regular orbits whose exceptional fibres over the origin reveal similar properties. We construct a morphism between these two resolutions, contracting exactly the excess part of the exceptional fibre. This construction is motivated by the study of some pencils of K3 surfaces appearing as minimal resolutions...