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.
The first author is supported by the European Research Council, project number 306445.
The second author is partially supported by an NSF grant.
"The floating body in real space forms." J. Differential Geom. 110 (2) 187 - 220, October 2018. https://doi.org/10.4310/jdg/1538791243