Frankfurt, 26 September 2011.
Abstract Classical integral geometry takes place in Euclidean space, but one can attempt to imitate it in any other metric space. In particular, one can attempt this in Rn equipped with the metric derived from the p-norm. This has been thoroughly investigated for 1 < p < ∞, but not for p = 1 .
Integral geometry for the 1-norm (in the metric sense above) bears a striking resemblance to integral geometry for the 2-norm, but is radically different from that for all other values of p. I will give a Hadwiger-type theorem for Rn with the 1-norm, and analogues of the classical formulas of Steiner, Crofton and Kubota. I will also give principal and higher kinematic formulas. Each of these results is closely analogous to its Euclidean counterpart, yet the proofs are quite different.
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Paper Integral geometry for the 1-norm, arXiv:1012.5881