82 research outputs found
A multiplicative characterization of the power means
A startlingly simple characterization of the p-norms has recently been found
by Aubrun and Nechita (arXiv:1102.2618) and by Fernandez-Gonzalez, Palazuelos
and Perez-Garcia. We deduce a simple characterization of the power means of
order greater than or equal to 1.Comment: 7 pages. Version 3: references added; minor edit
Generalized enrichment of categories
We define the phrase `category enriched in an fc-multicategory' and explore
some examples. An fc-multicategory is a very general kind of 2-dimensional
structure, special cases of which are double categories, bicategories, monoidal
categories and ordinary multicategories. Enrichment in an fc-multicategory
extends the (more or less well-known) theories of enrichment in a monoidal
category, in a bicategory, and in a multicategory. Moreover, fc-multicategories
provide a natural setting for the bimodules construction, traditionally
performed on suitably cocomplete bicategories. Although this paper is
elementary and self-contained, we also explain why, from one point of view,
fc-multicategories are the natural structures in which to enrich categories.Comment: 18 pages; written 199
Integral geometry for the 1-norm
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.Comment: 17 pages. Version 3: minor clarifications. This version will appear
in Advances in Applied Mathematic
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