  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 R^n equipped with the metric derived from the p-norm. This has, in
effect, been investigated intensively for 1<p<\infty, but not for p=1. We show
that integral geometry for the 1-norm bears a striking resemblance to integral
geometry for the 2-norm, but is radically different from that for all other
values of p. We prove a Hadwiger-type theorem for R^n with the 1-norm, and
analogues of the classical formulas of Steiner, Crofton and Kubota. We also
prove principal and higher kinematic formulas. Each of these results is closely
analogous to its Euclidean counterpart, yet the proofs are quite different.
