Mathematics – Functional Analysis
Scientific paper
2010-08-20
Mathematics
Functional Analysis
14 pages, 2 figures, 2 tables
Scientific paper
Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable metric trees that have generalized roundness precisely one. At the outset we consider spherically symmetric trees endowed with the usual combinatorial metric (SSTs). Using a simple geometric argument we show how to determine decent upper bounds on the generalized roundness of finite SSTs that depend only on the downward degree sequence of the tree in question. By considering limits it follows that if the downward degree sequence $(d_{0}, d_{1}, d_{2}...)$ of a SST $(T,\rho)$ satisfies $|\{j \, | \, d_{j} > 1 \}| = \aleph_{0}$, then $(T,\rho)$ has generalized roundness one. Included among the trees that satisfy this condition are all complete $n$-ary trees of depth $\infty$ ($n \geq 2$), all $k$-regular trees ($k \geq 3$) and inductive limits of Cantor trees. The remainder of the paper deals with two classes of countable metric trees of generalized roundness one whose members are not, in general, spherically symmetric. The first such class of trees are merely required to spread out at a sufficient rate (with a restriction on the number of leaves) and the second such class of trees resemble infinite combs.
Caffarelli Elena
Doust Ian
Weston Anthony
No associations
LandOfFree
Metric trees of generalized roundness one does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Metric trees of generalized roundness one, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Metric trees of generalized roundness one will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-80306