Hello,
The next Banach spaces webinar is on Friday April 30 at 9AM Central time. Please join us at
https://unt.zoom.us/j/83807914306
Speaker: Beata Randrianantoanina (Miami University in Ohio) Title: On $L_1$-embeddability of unions of $L_1$-embeddable metric spaces and of twisted unions of hypercubes
Abstract: Let $\mathcal{E}$ be a class of metric spaces, $(X,d)$ be a metric space, and $A,B$ be metric subspaces of $X$ such that $X=A\cup B$ and $(A,d), (B,d)$ embed bilipschitzly into spaces $E_A,E_B\in \mathcal{E}$ with distortions $D_A, D_B$, respectively. Does this imply that there exists a constant $D$ depending only on $D_A, D_B$, and the class $\mathcal{E}$, so that $(X,d)$ embeds bilipschitzly into some space $E_X\in \mathcal{E}$ with distortion $D$?
This question was answered affirmatively for the class $\mathcal{E}$ of all ultrametric spaces by Mendel and Naor in 2013, and for the class $\mathcal{E}$ of all Hilbert spaces by K. Makarychev and Y. Makarychev in 2016. K. Makarychev and Y. Makarychev in 2016 conjectured that the answer is negative when $\mathcal{E}$ is a class of $\ell_p$-spaces for any fixed $p\notin{2,\infty},$ in particular for $p=1$. In this connection, Naor in 2015 and Naor and Rabani in 2017 asked whether the metric space known as ``twisted union of hypercubes'', first introduced by Lindenstrauss in 1964, and also considered by Johnson and Lindenstrauss in 1986, embeds into $\ell_1$.
In this talk I will show how to embed general classes of twisted unions of $L_1$-embeddable metric spaces into $\ell_1$, including twisted unions of hypercubes whose metrics are determined by concave functions of the $\ell_1$-norm, and discuss some related results (joint work with Mikhail I. Ostrovskii).
For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach
Thank you, and best regards,
Bunyamin Sari