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Alexandrov’s isodiametric conjecture and the cut locus of a surface. (English) Zbl 1341.52012

Posed by A. D. Alexandrow [Die innere Geometrie der konvexen Flächen. Berlin: Akademie-Verlag (1955; Zbl 0065.15102)], the isodiametric conjecture claims that, for any closed, oriented, connected, convex and smooth \(2\)-dimensional Riemannian manifold with area \(A\) and intrinsic diameter D, we have \[ \frac{A}{D^2} \leq \frac{\pi}{2}. \] Although settled for some special cases, the conjecture, in its full generality, is still open. The authors prove that, if the manifold has a point \(p\) for which the cut locus \({C}_p\) is countable, then \[ \frac{A}{D^2} \leq \frac{\pi}{2} + \frac{|{C}_p|}{\rho}, \] where \(| {C}_p|\) is the total length of the cut locus, which is known to have Hausdorff dimension \(1\), and \(\rho\) is the injectivity radius of the manifold. They are thus validating the conjecture in the case of a manifold which admits a point \(p\) whose cut locus consists of a single point, as for example ellipsoids.
The proof relies on a symmetrization procedure which transforms the given \(2\)-dimensional manifold into a surface of revolution with a ratio larger or equal than \(A/D^2\).

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
53A05 Surfaces in Euclidean and related spaces
53C22 Geodesics in global differential geometry
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
52A15 Convex sets in \(3\) dimensions (including convex surfaces)

Citations:

Zbl 0065.15102

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Full Text: arXiv Euclid

References:

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