Integral geometry of pairs of hyperplanes or lines
Daniel Hug, Rolf Schneider
TL;DR
This paper explores integral geometry measures for pairs of hyperplanes or lines intersecting convex bodies, revealing simplified results when the bodies have constant width or brightness.
Contribution
It introduces motion invariant measures for pairs of hyperplanes or lines intersecting convex bodies, with special cases for bodies of constant width or brightness.
Findings
Simplified measures for bodies of constant width or brightness
Characterization of intersection conditions for pairs of hyperplanes or lines
Extension of Crofton's formula to pairs of geometric objects
Abstract
Crofton's formula of integral geometry evaluates the total motion invariant measure of the set of -dimensional planes having nonempty intersection with a given convex body. This note deals with motion invariant measures on sets of pairs of hyperplanes or lines meeting a convex body. Particularly simple results are obtained if, and only if, the given body is of constant width in the first case, and of constant brightness in the second case.
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