For R-ball convex bodies, the volume lost by the R-ball floating body at scale δ behaves like δ^{2/(n+1)} times an integral of (κ_i - 1/R)^{1/(n+1)} over the boundary, defining a rigid-motion invariant valuation.
General Higher Order $L^p$ Mean Zonoids
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abstract
In 1970, Schneider introduced the higher-order difference body and the associated Rogers-Shephard inequality. Recently, Haddad, Langharst, Putterman, Roysdon and Ye expanded the concept to a burgeoning higher-order Brunn-Minkowski theory. In 1991, Zhang introduced mean zonoids of a convex body, which was extended to the Firey-Brunn-Minkowski theory setting by Xi, Guo and Leng in 2014. In this note, we extend these $L^p$ mean zonoids to the higher-order setting and establish the associated isoperimetric inequality.
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Floating bodies for ball-convex bodies
For R-ball convex bodies, the volume lost by the R-ball floating body at scale δ behaves like δ^{2/(n+1)} times an integral of (κ_i - 1/R)^{1/(n+1)} over the boundary, defining a rigid-motion invariant valuation.