Morse Quasiflats II

Huang, JingYin; Kleiner, Bruce; Stadler, Stephan

doi:10.48550/arxiv.2003.08912

Cited by 3 publications

(5 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In practice, it is often desirable to impose stronger properties on a bicombing than (1.1). See [9,25,29] for some recent examples. By assuming that a conical bicombing is consistent, see Section 2.3 for the definition, one obtains an interesting class of bicombings that seem to be quite rigid.…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

A non-compact convex hull in general non-positive curvature

Basso¹,

Krifka²

2023

Preprint

View full text Add to dashboard Cite

In this article, we are interested in metric spaces that satisfy a weak non-positive curvature condition in the sense that they admit a conical bicombing. Recently, these spaces have begun to be studied in more detail, and a rich theory is beginning to emerge. We contribute to this study by constructing a complete metric space X with a conical bicombing σ such that there is a finite subset of X whose closed σ-convex hull is non-compact. In CAT(0)-geometry, the analogous statement is an open question, i.e. it is not known whether closed convex hulls of finite subsets of complete CAT(0) space are compact or not. This question goes back to Gromov. Our result shows that to obtain a positive answer to Gromov's question, more than just the convexity properties of the metric must be used. The constructed space X has the additional property that there is an integer n such that it is an initial object in the category of convex hulls of n-point sets. Thus, roughly speaking, X can be thought of as the largest possible convex hull of n points.

show abstract

Section: Proof Of Main Resultsmentioning

confidence: 99%

A non-compact convex hull in general non-positive curvature

Basso¹,

Krifka²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…The stronger version of the coning inequality asserted in the corollary is called strong coning inequality in [20] and [21] and plays an important role in these papers.…”

Section: Proof Of Theorem 11 and Generalizationsmentioning

confidence: 99%

“…This result was later generalized to complete metric spaces by the second named author in [38]. Coning inequalities have also played a crucial role in recent articles on higher rank hyperbolicity [24], Morse quasiflats [20], [21], and the equivalence of flat and weak convergence of currents [39]. It is open in general whether spaces with k-dimensional Euclidean isoperimetric inequalities admit k-dimensional coning inequalities.…”

mentioning

confidence: 97%

“…Using a suitable variant of Theorem 1.1 established in Section 5.1 we will actually get a stronger version of the coning inequality in which we also obtain control over the diameter of the filling chain; see Corollary 5.2. Coning inequalities play an important role in the recent articles [20], [21], [24], and the stronger version of the coning inequality established in Corollary 5.2 prominently appears in [20], [21], where it is called strong coning inequality.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Undistorted fillings in subsets of metric spaces

Basso¹,

Wenger²,

Young³

2021

Preprint

View full text Add to dashboard Cite

We prove that if a quasiconvex subset X of a metric space Y has finite Nagata dimension and is Lipschitz k-connected or admits Euclidean isoperimetric inequalities up to dimension k for some k then X is isoperimetrically undistorted in Y up to dimension k + 1. This generalizes and strengthens a recent result of the third named author and has several consequences and applications. It yields for example that in spaces of finite Nagata dimension, Lipschitz connectedness implies Euclidean isoperimetric inequalities, and Euclidean isoperimetric inequalities imply coning inequalities. It furthermore allows us to prove an analog of the Federer-Fleming deformation theorem in spaces of finite Nagata dimension admitting Euclidean isoperimetric inequalities.

show abstract

“…In [30], Kleiner introduced often convex metric spaces which in our terminology are metric spaces with a consistent convex bicombing. We refer to [31,6,23] for some recent applications of consistent convex bicombings.…”

Section: Improving Conical Bicombingsmentioning

confidence: 99%