Abstract.
We prove that sets of positive reach in Riemannian manifolds and more generally, almost convex subsets in spaces with an upper curvature bound have an upper curvature bound with respect to the inner metric.
References
Alexander, S., Bishop, -convex functions on metric spaces. Manuscripta Math. 110, 115–133 (2003)
Alexander, S., Berg, I., Bishop, R.: Geometric curvature bounds in Riemannian manifolds with boundary. Trans. Am. Math. Soc. 339, 703–716 (1993)
Aleksandrov, A.D.: Ruled surfaces in metric spaces. Vestnik Leningrad. Univ. 12, 5–26 (1957)
Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry. Graduate Studies in Mathematics 33, 2001
Bridson, M., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften, 1999
Federer, H.: Curvature measures. Trans. Amer. Math. Soc. 93, 418–491 (1959)
Lytchak, A.: Almost convex subsets. Preprint
Petrunin, A.: Metric minimizing surfaces. Electron. Res. Announc. Amer. Math. Soc. 5, 47–54 (1999)
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Mathematics Subject Classification (2000): 53C20
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Lytchak, A. On the geometry of subsets of positive reach. manuscripta math. 115, 199–205 (2004). https://doi.org/10.1007/s00229-004-0491-8
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DOI: https://doi.org/10.1007/s00229-004-0491-8