Testing for Convexity

Kostya Rybnikov

UMass Lowell

April 2,
refreshments at 3:45pm


A hypersurface M immersed in R^n (S^n, H^n, or another manifold with an affine connection) is locally convex if each point of M has an M-neighborhood which lies on the boundary of a convex body. Under what additional conditions does this guarantee that M, as a whole, is the boundary of a convex body?

This type of questions has been addressed at the end of the 19th century by Hadamard, E. Schmidt in the 1920s, A.D. Aleksandrov and van Heijenoort in the 1940-50s, Jonker & Norman, V. Arnol'd in the early 1970s, and more recently by Mehlhorn et al., Preparata et al., and the speaker.

We will present the strongest known criterion for global convexity for PL-surfaces in R^n and S^n. Recent results and conjectures for general surfaces will also be discussed. Some of the material for the PL-surfaces (but not all, and only for the PL-case) can be found at math.MG/0309370 on ArXiv.org .

Speaker's Contact Info: krybniko(at-sign)cs.uml.edu

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Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)math.mit.edu

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