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Integral representations of mixed volumes

Banff International Research Station for Mathematical Innovation and Discovery

The notion of mixed volumes $V(K_1,\dots, K_d)$ of convex bodies $K_1,\dots ,K_d$ in Euclidean space $\rd$ is of central importance in the Brunn-Minkowski theory. Representations for mixed volumes are available in special cases, for example as integrals over the unit sphere with respect to mixed area measures. More generally, in Hug-Rataj-Weil (2013) a formula for $V(K [n], M[d-n]), n\in \{1,\dots ,d-1\},$ as a double integral over flag manifolds was established which involved certain flag measures of the convex bodies $K$ and $M$ (and required a general position of the bodies). In the talk, we discuss the general case $V(K_1[n_1],\dots , K_k[n_k]), n_1+\cdots +n_k=d,$ and show a corresponding result involving the flag measures $\Omega_{n_1}(K_1;\cdot),\dots, \Omega_{n_k}(K_k;\cdot)$. For this purpose, we first establish a curvature representation of mixed volumes over the normal bundles of the bodies involved. We also point out a connection of the latter result to a combinatorial formula of R. Schneider, in the case of polytopes. Joint work with Daniel Hug (Karlsruhe) and Jan Rataj (Prague).

Mathematics; Convex and discrete geometry; Geometry; Discrete mathematics

video/mp4

Author affiliation: Karlsruhe Institute of Technology

2017-11-21

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63673

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/