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.
Andrews, Gauss curvature flow: the fate of rolling stone , Invent
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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.