TY - JOUR

T1 - Self-contracted curves in spaces with weak lower curvature bound

AU - Lebedeva, Nina

AU - Ohta, Shin-ichi

AU - Zolotov, Vladimir

N1 - A goofy mistake in formulations of Theorem 2 and 4 is fixed

PY - 2021/6

Y1 - 2021/6

N2 - We show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound in a sense we introduce in this article. This class of spaces is wide and includes, for example, finite-dimensional Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative flag curvature. (To be more precise, our condition is regarded as a strengthened doubling condition and holds also for a certain class of metric spaces with upper curvature bound.) We also provide the non-embeddability of large snowflakes into (balls in) metric spaces in the same class. We follow the strategy of the last author's previous paper based on the small rough angle condition, where spaces with upper curvature bound are considered. The results in this article show that such a strategy applies to spaces with lower curvature bound as well.

AB - We show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound in a sense we introduce in this article. This class of spaces is wide and includes, for example, finite-dimensional Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative flag curvature. (To be more precise, our condition is regarded as a strengthened doubling condition and holds also for a certain class of metric spaces with upper curvature bound.) We also provide the non-embeddability of large snowflakes into (balls in) metric spaces in the same class. We follow the strategy of the last author's previous paper based on the small rough angle condition, where spaces with upper curvature bound are considered. The results in this article show that such a strategy applies to spaces with lower curvature bound as well.

KW - math.MG

KW - 51F99

KW - Snowflake

KW - Self-contracted curve

KW - Isometric embedding

UR - https://arxiv.org/pdf/1902.01594.pdf

UR - http://www4.math.sci.osaka-u.ac.jp/~sohta/papers/LOZ.pdf

UR - https://www.mendeley.com/catalogue/53700ec6-66da-3f5f-aa0e-2fa9a777981b/

U2 - 10.1093/imrn/rnz347

DO - 10.1093/imrn/rnz347

M3 - Article

VL - 2021

SP - 8623

EP - 8656

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 11

ER -