2010/08/31 by Elena Caffarelli, Elena A. Caffarelli, Ian Doust +1 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Combinatorics #Countable set #Discrete mathematics #Geometry #Graph theory and applications #Mathematics #Metric space #Point processes and geometric inequalities #Roundness (object) #Tree (set theory) #math.FA #msc:46B20
paper · pdf · doi:10.1007/s00010-011-0108-8
published as Aequationes Math., 83 (2012), 239-235 · 14 pages, 2 figures, 2 tables
arxiv created 2011/05/31 · openalex publication_date 2011/12/26 · arxiv updated 2016/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 (d0, d1, d2...) of a SST (T,ρ) satisfies |\j | dj > 1 \| = ℵ0, then (T,ρ) has generalized roundness one. Included among the trees that satisfy this condition are all complete n-ary trees of depth ∞ (n ≥ 2), all k-regular trees (k ≥ 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.