Hedgehogs and Zonoids

✍ Scribed by Yves Martinez-Maure

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 145 KB
• Volume: 158
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.2000.1943

✦ Synopsis

Let R be a continuous real function on the unit sphere S n of (n+1)-dimensional Euclidean space R n+1 . We prove that the map

where ( } , } ) is the standard inner product and _ the spherical Lebesgue measure, is of class C 2 . It follows that the boundaries of zonoids (resp. generalized zonoids), whose generating measure have a continuous density with respect to _, can be considered as hedgehogs (envelopes parametrized by their Gauss map) with a C 2 support function. We deduce a local property for such zonoids. We give a formula for the curvature function of the hedgehog defined by h and we deduce a necessary and sufficient condition for h being the support function of a convex body of class C 2 + . We define projection hedgehogs (resp. mixed projection hedgehogs) and interpret their support functions in terms of n-dimensional volume (resp. mixed volume). Finally, we consider the extension of the classical Minkowski problem to hedgehogs.