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Unduloids and their geometry

Mariana Hadzhilazova, Ivaïlo M. Mladenov, John Oprea (2007)

Archivum Mathematicum

In this paper we consider non-compact cylinder-like surfaces called unduloids and study some aspects of their geometry. In particular, making use of a Kenmotsu-type representation of these surfaces, we derive explicit formulas for the lengths and areas of arbitrary segments, along with a formula for the volumes enclosed by them.

Uniqueness results for the Minkowski problem extended to hedgehogs

Yves Martinez-Maure (2012)

Open Mathematics

The classical Minkowski problem has a natural extension to hedgehogs, that is to Minkowski differences of closed convex hypersurfaces. This extended Minkowski problem is much more difficult since it essentially boils down to the question of solutions of certain Monge-Ampère equations of mixed type on the unit sphere 𝕊 n of ℝn+1. In this paper, we mainly consider the uniqueness question and give first results.

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