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Curvature Relations and Affine Surface Area for a General Convex Body and its Polar

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Abstract

We investigate the relationship between generalized curvatures of an arbitrary convex body K and its polar body K* in d-dimensional Euclidean space. For example, the generalized Gauß-Kronecker curvature of K is compared with the product of the generalized principal radii of curvature of K*. This leads to a generalization of the classical statement saying that the product of the equiaffine support functions of K and K* is equal to 1, provided K is sufficiently smooth and has positive Gauß-Kronecker curvature. Another consequence concerns the equality of the extended p-affine surface area of K and the q-affine surface area of K*, if pq = d2. In the special case of a smooth convex body and for p = d this result is well known in centroaffine differential geometry.

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  1. Mathematisches Institut, Albert-Ludwigs-Universität, Albertstraße 23b, D-79104, Freiburg i. Br, Germany

    Daniel Hug

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  1. Daniel Hug
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Correspondence to Daniel Hug.

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Hug, D. Curvature Relations and Affine Surface Area for a General Convex Body and its Polar. Results. Math. 29, 233–248 (1996). https://doi.org/10.1007/BF03322221

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