Journal of Differential Geometry

The floating body in real space forms

Florian Besau and Elisabeth M. Werner

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Abstract

We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space.

Our main result establishes a relation between the derivative of the volume of the floating body and a certain surface area measure, which we called the floating area. In the Euclidean setting the floating area coincides with the well known affine surface area, a powerful tool in the affine geometry of convex bodies.

Note

The first author is supported by the European Research Council, project number 306445.

Note

The second author is partially supported by an NSF grant.

Article information

Source
J. Differential Geom., Volume 110, Number 2 (2018), 187-220.

Dates
Received: 26 April 2016
First available in Project Euclid: 6 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1538791243

Digital Object Identifier
doi:10.4310/jdg/1538791243

Mathematical Reviews number (MathSciNet)
MR3861810

Zentralblatt MATH identifier
06958640

Subjects
Primary: 52A55: Spherical and hyperbolic convexity
Secondary: 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 53A35: Non-Euclidean differential geometry

Keywords
hyperbolic convex geometry hyperbolic floating body affine surface area

Citation

Besau, Florian; Werner, Elisabeth M. The floating body in real space forms. J. Differential Geom. 110 (2018), no. 2, 187--220. doi:10.4310/jdg/1538791243. https://projecteuclid.org/euclid.jdg/1538791243


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