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## Studia Mathematica

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## Diversity-normed spaces and diversity embeddings

### Tom 267 / 2022

Studia Mathematica 267 (2022), 19-35 MSC: Primary 46B04; Secondary 46B85, 30L05. DOI: 10.4064/sm210629-7-1 Opublikowany online: 10 June 2022

#### Streszczenie

The purpose of this paper is to extend some known metric embedding results to the setting of diversities introduced in [D. Bryant, P. F. Tupper, Adv. Math. 231 (2012), 3172–3198]. We first introduce diversity-normed spaces as a generalization of normed spaces and investigate their relationships with diversities. In particular, we introduce $L_p$-diversity-normed spaces $(1\leq p \leq \infty )$ which can be simultaneously considered as ($L_p$-)diversities. Then, for any $p$ $(1 \leq p \leq \infty )$, we investigate the possibility of embedding finite diversities and ultradiversities into $L_p$-diversities $\ell _p^d$, for some positive integer $d$, with some distortion. We present results analogous to the Bourgain theorem in the setting of both diversities and ultradiversities. We show that every diversity on $n$ points embeds in the diversities: (i) $\ell _p^{\mathcal {O}( \log n)}$ $(1\leq p \leq 2)$ with distortion $\mathcal {O}(n \log ^{{(1+p)}/{p}}n)$; (ii) $\ell _p^{\mathcal {O}(\log ^{2}n)}$ $(2 \lt p \lt \infty )$ with distortion $\mathcal {O}(n \log ^{ {(2+p)}/{p}}n)$; (iii) $\ell _\infty ^{\mathcal {O}(\log ^2n)}$ with distortion $\mathcal {O}(n\log n)$. In addition, each ultradiversity embeds in the diversity $\ell _p^{\mathcal {O}( \log n)}$ $(1 \leq p \lt \infty )$ with distortion $\mathcal {O}(\log ^{ {1}/{p}}n)$ as well as in $\ell _\infty ^{\mathcal {O}( \log n)}$ with constant distortion.

#### Autorzy

• Pouya HaghmaramFaculty of Mathematics
K. N. Toosi University of Technology
Tehran, Iran
e-mail
• Shohreh Golpaigani FardFaculty of Mathematics
K. N. Toosi University of Technology
Tehran, Iran
e-mail
• Kourosh NourouziFaculty of Mathematics
K. N. Toosi University of Technology
Tehran, Iran
e-mail

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