Abstract
Let K be a convex body in \(\mathbb{R}^n\) and let \(f : \partial K \to \mathbb{R}_ + \) be a continuous, positive function with \(\int_{\partial K} f(x )d\mu_{\partial K} (x ) = 1\) where \(\mu_{\partial K}\) is the surface measure on \(\partial K\). Let \(\mathbb{P}_f\) be the probability measure on \(\partial K\) given by \({\rm d}\mathbb{P}_f(x ) = f (x ){\rm d} \mu_{\partial K} (x )\). Let \(\kappa\) be the (generalized) Gauß-Kronecker curvature and \(\mathbb{E}(f,N )\) the expected volume of the convex hull of N points chosen randomly on \(\partial K\) with respect to \(\mathbb{P}_f\). Then, under some regularity conditions on the boundary of K
where c n is a constant depending on the dimension n only.
The minimum at the right-hand side is attained for the normalized affine surface area measure with density
Carsten Schütt: Partially supported by the Schrödinger Institute, Vienna, and by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, Jerusalem, sponsored by the Minerva Foundation (Germany).
Elisabeth Werner: Partially supported by the Schrödinger Institute, Vienna, by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, Jerusalem, sponsored by the Minerva Foundation (Germany) and by a grant from the National Science Foundation.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Editor information
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin/Heidelberg
About this chapter
Cite this chapter
Schütt, C., Werner, E. (2003). Polytopes with Vertices Chosen Randomly from the Boundary of a Convex Body. In: Milman, V.D., Schechtman, G. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 1807. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-36428-3_19
Download citation
DOI: https://doi.org/10.1007/978-3-540-36428-3_19
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00485-1
Online ISBN: 978-3-540-36428-3
eBook Packages: Springer Book Archive