需六个坐标实现等距ℓ_∞嵌入的32叶树
A 32-leaf tree requiring six coordinates for an isometric $\ell_\infty$ embedding
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中文总结 AI 辅助
该研究推翻了树的等距ℓ_∞嵌入维数相关猜想，构造出32叶树其最小等距ℓ_∞维数为6，证明≤31叶树符合原猜想并回答了相关学者的问题。
中文摘要 AI 辅助
我们推翻了“每棵有t片叶子的树都能等距嵌入ℓ_∞^⌈log₂t⌉”的猜想。构造了一棵有32片叶子的树，其最小等距ℓ_∞维数为6而非5，并证明每片叶子数不超过31的树都能达到该猜想的界限；Brigham等人已记录到21片叶子时仍符合该等式。因此32是首个失效的情况，该实例肯定地回答了Fitzpatrick和Nowakowski在2000年提出的一个问题。该拓扑结构在所有正边长分配下的维数均为6，因此也推翻了带权度量树的后续尖锐叶阈值猜想。
英文摘要
We disprove the conjecture that every tree with t leaves embeds isometrically into $\ell_\infty^{\lceil \log_2 t\rceil}$. We construct a 32-leaf tree whose least isometric $\ell_\infty$-dimension is six rather than five, and prove that every tree with at most 31 leaves attains the conjectured bound; Brigham et al. had recorded equality through 21 leaves. Thus 32 is the first failure, and the example answers affirmatively a question of Fitzpatrick and Nowakowski from 2000. The same topology has dimension six under every assignment of positive edge lengths, and therefore also disproves the later sharp leaf-threshold conjecture for weighted metric trees.
