Two important closely related notions in affine convex geometry are the floating
body and the affine surface area of a convex body. The floating body of a convex body is
obtained by cutting off caps of volume less or equal to a fixed positive constant. Taking
the right-derivative of the volume of the floating body gives rise to an affine invariant, the
affine surface area. This was established for all convex bodies in all dimensions by Schuett
and Werner. There is a natural inequality associated with affine surface area, the affine
isoperimetric inequality, which states that among all convex bodies, with fixed volume, affine
surface area is maximized for ellipsoids.
Due to its important properties, which make them effective and powerful tools, affine
surface area and floating body are omnipresent in geometry and have applications in many
other areas of mathematics, e.g., in problems of approximation of convex bodies by polytopes
and for the notion of halfspace depth for multivariate data from statistics.
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