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Crofton formulas on simply connected Riemannian space forms allow the volumes, or more generally the Lipschitz–Killing curvature integrals of a submanifold with corners, to be computed by integrating the Euler characteristic of its intersection with all geodesic submanifolds. We develop a framework of Crofton formulas with distributions replacing measures, which has in its core Alesker's Radon transform on valuations. We then apply this framework, and our recent Hadwiger-type classification, to compute explicit Crofton formulas for all isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and pseudohyperbolic spaces. We find that, in essence, a single measure which depends analytically on the metric, gives rise to all those Crofton formulas through its distributional boundary values at parts of the boundary corresponding to the different indefinite signatures. In particular, the Crofton formulas we obtain are formally independent of signature.
The paper studies the relation between the asymptotic values of the ratios area/length (F/L) and
diameter/length (D/L) of a sequence of convex sets expanding over the whole hyperbolic plane. It is known
that F/L goes to a value between 0 and 1 depending on the shape of the contour. In the paper, it is first of
all seen that D/L has limit value between 0 and 1/2 in strong contrast with the euclidean situation in which
the lower bound is 1/π (D/L = 1/π if and only if the convex set has constant width). Moreover, it is
shown that, as the limit of D/L approaches 1/2, the possible limit values of F/L reduce. Examples of all
possible limits F/L and D/L are given.
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