1-范数的Integral geometry
Integral geometry for the 1-norm
摘要：
经典的积分几何发生在欧几里得空间，但人们可以尝试在任何其他度量空间中模仿它。特别地，可以在R^n中尝试这一点，并配备从p范数导出的度量。实际上，这已经针对1<p< \infty而不是p=1进行了深入研究。我们证明了1-范数的积分几何与2-范数的积分几何有着惊人的相似之处，但与p的所有其他值的积分几何有着根本的不同。我们证明了R^n具有1-范数的Hadwiger型定理，以及Steiner，Crofton和Kubota的经典公式的类似物。我们还证明了主要的和更高的运动公式。这些结果中的每一个都非常类似于它的欧几里德对应物，但证明是完全不同的。
英文摘要：
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.
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