Abstract
We mainly prove that most d-dimensional convex surfaces \(\Sigma \) have a set of endpoints of Hausdorff dimension at least \(d-2\). A recent previous work of the author gave d / 3 as a lower bound, which is thus improved when \(d \ge 4\). Our proof uses a rather simple example where this dimension is \(d-2\) and the corresponding Hausdorff measure is positive. We also address several related problems. An endpoint is a point not lying in the interior of any shortest path in \(\Sigma \). “Most” means that the exceptions constitute a meager set, relatively to the usual Pompeiu-Hausdorff distance.
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Notes
- 1.
One can also use as a more explicit denomination the term “Euclidean closed convex hypersurface”.
- 2.
- 3.
The set of outer unit normal vectors to \(\Sigma \) at points of \(U_\infty (C)\).
- 4.
- 5.
The idea of considering that kind of exemple came rather lately to my mind, after a conversation with David Guy.
- 6.
The tangent cone \(T_p(\Sigma )\) defined in the introduction is known to be isometric to the abstract tangent cone used in the framework of Alexandrov surfaces.
- 7.
We do not know a direct proof of Lemma 1, though it should exist for a so simple set.
- 8.
Indeed the unit ball of \(T_x\Sigma \) has smaller \(\mathcal {H}^d\)-measure than the unit ball of
, the ratio being \(\displaystyle 2^{q-1} \sin ^{q-1} {\arctan r \over 2}\). - 9.
True for example in finite dimension.
- 10.
A proof of this is given in Lemma 4 of [15].
, the ratio being References
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Acknowledgments
Thanks are due to Costin Vîlcu for many informations and useful suggestions, and to the referee, who greatly helped to improve the English and the presentation, and to eliminate some mistakes.
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Rivière, A. (2016). About the Hausdorff Dimension of the Set of Endpoints of Convex Surfaces. In: Adiprasito, K., Bárány, I., Vilcu, C. (eds) Convexity and Discrete Geometry Including Graph Theory. Springer Proceedings in Mathematics & Statistics, vol 148. Springer, Cham. https://doi.org/10.1007/978-3-319-28186-5_8
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