\documentclass{amsart} % options include 12pt or 11pt or 10pt % classes include article, report, book, letter, thesis \usepackage{setspace} \usepackage{a4} \usepackage{amsthm} \usepackage{latexsym} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{textcomp} \usepackage{cite} \usepackage{enumerate} \usepackage{amssymb} \usepackage{hyperref} \usepackage{amsmath} \usepackage{tikz} \usepackage[mathscr]{euscript} \usepackage{mathtools} \newtheorem{theorem}{Theorem}[section] \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem] {Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{lemma} [theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{observation}[theorem]{Observation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{question}[theorem]{Question} \setlength{\parindent}{0pt} \setlength{\evensidemargin}{0.3cm} \setlength{\oddsidemargin}{0.3cm} \setlength{\topmargin}{-1cm} \textwidth 16cm \textheight 23cm \onehalfspacing \title{This is the title} \newcommand{\Cross}{\mathbin{\tikz [x=1.4ex,y=1.4ex,line width=.2ex] \draw (0,0) -- (1,1) (0,1) -- (1,0);}}% \raggedbottom \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\blankbox}[2]{% \parbox{\columnwidth}{\centering \setlength{\fboxsep}{0pt}% \fbox{\raisebox{0pt}[#2]{\hspace{#1}}}% }% } \usepackage{fancyhdr} %\pagestyle{fancy} \pagestyle{fancy} %\thispagestyle{empty} \fancyhead[LO]{\textbf{Ultrametric Bourgain-Figiel-Milman, Enflo Type and Mendel-Naor Cotype Problems}} \fancyhead[RE]{\textbf{K. MAHESH KRISHNA}} \begin{document} \hrule\hrule\hrule\hrule\hrule \vspace{0.3cm} \begin{center} {\bf{Ultrametric Bourgain-Figiel-Milman, Enflo Type and Mendel-Naor Cotype Problems}}\\ \vspace{0.3cm} \hrule\hrule\hrule\hrule\hrule \vspace{0.3cm} \textbf{K. MAHESH KRISHNA}\\ School of Mathematics and Natural Sciences\\ Chanakya University Global Campus\\ NH-648, Haraluru Village\\ Devanahalli Taluk, Bengaluru North District\\ Karnataka State, 562 110, India\\ Email: kmaheshak@gmail.com \\ Date: \today \end{center} \hrule\hrule \vspace{0.5cm} %-------------------------------------- \textbf{Abstract}: We ask for ultrametric version of following three: (1) Bourgain-Figiel-Milman Theorem, (2) Enflo Type, (3) Mendel-Naor Cotype. \textbf{Keywords}: Bourgain-Figiel-Milman Theorem, Enflo Type, Mendel-Naor Cotype, Ultrametric space. \textbf{Mathematics Subject Classification (2020)}: 47B10, 47L20, 26E30, 46S10, 47S10, 12J25. \hrule %\tableofcontents \hrule \section{Ultrametric Bourgain-Figiel-Milman Problem} Let $\mathcal{X}$ and $\mathcal{Y}$ be finite dimensional normed linear spaces such that $\dim (\mathcal{X})=\dim(\mathcal{Y})$. The \textbf{Banach-Mazur distance} between $\mathcal{X}$ and $\mathcal{Y}$ is defined as \begin{align*} d_{BM}(\mathcal{X}, \mathcal{Y})\coloneqq \inf\{\|T\|\|T^{-1}\|: T: \mathcal{X} \to \mathcal{Y} \text{ is invertible linear operator}\}. \end{align*} For $n \in \mathbb{N}$, let $(\mathbb{R}^n, \langle \cdot, \cdot \rangle)$ be the standard Euclidean Hilbert space. Following wonderful result says that every finite dimensional Banach space has a subspace which is close (in the Banach-Mazur distance) to the Euclidean space. \begin{theorem}\cite{DVORETZKY, ALBIACKALTON} \label{DA}\textbf{(Dvoretzky-Milman Theorem)} There is a universal constant $C>0$ satisfying the following property: If $\mathcal{X}$ is any $n$-dimensional real Banach space and $0<\varepsilon < \frac{1}{3}$, then for every natural number \begin{align*} k\leq C \frac{\varepsilon^2}{|\log \varepsilon|} \log n, \end{align*} there exists a $k$-dimensional Banach subspace $\mathcal{Y}$ of $\mathcal{X}$ such that \begin{align*} d_{BM}(\mathcal{Y}, (\mathbb{R}^k, \langle \cdot, \cdot \rangle))<1+\varepsilon. \end{align*} \end{theorem} We refer \cite{MILMANSCHECHTMAN, MILMAM, MILMAN1992, FIGIEL2, SZANKOWSKI, MATOUSEK, MATSAKPLICHKO, PISIERBOOK, GORDONGUEDONMEYER, FIGIEL, FIGIEL3, FIGIELLINDENSTRAUSSMILMAN, SCHECHTMAN, GORDON2, PISIERBOOK2} for more information on Dvoretzky-Milman theorem. With Theorem \ref{DA}, we ask about the metric space version of that. Let $\mathcal{M}$ and $\mathcal{N}$ be finite metric spaces with $o(\mathcal{M})=o(\mathcal{N})$. Given a map $f: \mathcal{M} \to \mathcal{N}$, define \begin{align*} \|f\|_{\operatorname{Lip}}\coloneqq \sup\left\{\frac{d(f(x), f(y))}{d(x,y)}: x, y \in \mathcal{M}, x \neq y \right\}. \end{align*} The \textbf{Bourgain-Figiel-Milman distance} between $\mathcal{M}$ and $\mathcal{N}$ is defined as \begin{align*} d_{BFM}(\mathcal{M}, \mathcal{N})\coloneqq \inf\{\|f\|_{\operatorname{Lip}}\|f^{-1}\|_{\operatorname{Lip}}: f: \mathcal{M} \to \mathcal{N} \text{ is injective}\}. \end{align*} In 1986, Bourgain, Figiel and Milman proved the following nonlinear Dvoretzky theorem \cite{BOURGAINFIGIELMILMAN}. \begin{theorem}\cite{BOURGAINFIGIELMILMAN} \label{BFM}(\textbf{Bourgain-Figiel-Milman Theorem}) For every $\varepsilon >0$, there is a universal constant $C(\varepsilon)>0$ satisfying the following: Every finite metric space $\mathcal{M}$ contains a subset $\mathcal{N}$ satisfying following conditions. \begin{enumerate}[\upshape(i)] \item There is an injective map $f: \mathcal{N} \to \ell^2 (\mathbb{N}, \mathbb{R})$ such that \begin{align*} d_{BFM}(\mathcal{N}, f(\mathcal{N}))\leq 1+\varepsilon. \end{align*} \item \begin{align*} o(\mathcal{N})\geq C(\varepsilon)\log (o(\mathcal{M})). \end{align*} \end{enumerate} Further, following holds: If $\varepsilon <1$, then we can take \begin{align*} C(\varepsilon)=\frac{c_1\varepsilon}{\log (\frac{c_2}{\varepsilon})} \quad \text{ for some } c_1>0, c_2>0. \end{align*} \end{theorem} We will formulate problem based on Theorem \ref{BFM} to ultrametric spaces. \begin{definition} \cite{SCHIKHOF} Let $\mathcal{M}$ be a set. A map $d: \mathcal{M}\times \mathcal{M} \to [0, \infty)$ is said to be an \textbf{ ultrametric} if it satisfies following conditions. \begin{enumerate}[\upshape(i)] \item $d(z, z)=0$ for all $z \in \mathcal{M}$. If $x, y \in \mathcal{M}$ are such that $d(x, y)=0$, then $x=y$. \item $d(x, y)=d(y, x)$ for all $x, y \in \mathcal{M}$. \item $d(x, y)\leq \max\{d(x, z), d(z, y)\}$ for all $x, y, z \in \mathcal{M}$. \end{enumerate} In this case, $(\mathcal{M}, d)$ is called as an \textbf{ultrametric space}. \end{definition} Let $\mathbb{K}$ be a non-Archimedean valued field. We define \begin{align*} c_0 (\mathbb{N}, \mathbb{K})\coloneqq \left\{\{a_n\}_{n}: a_n \in \mathbb{K}, \forall n \in \mathbb{N}, \lim_{n \to \infty}a_n=0\right\} \end{align*} equipped with the ultra-norm \begin{align*} \|\{a_n\}_{n}\| \coloneqq \max_{n \in \mathbb{N}}|a_n|, \quad \forall \{a_n\}_{n} \in c_0 (\mathbb{N}, \mathbb{K}) \end{align*} and the p-adic inner product \begin{align*} \langle \{a_n\}_{n}, \{b_n\}_{n}\rangle \coloneqq \sum_{n=1}^{\infty}a_nb_n, \quad \forall \{a_n\}_{n}, \{b_n\}_n \in c_0 (\mathbb{N}, \mathbb{K}). \end{align*} Then $ c_0(\mathbb{N}, \mathbb{K})$ is a non-Archimedean Banach space which is also a p-adic Hilbert space \cite{KALISCH, GARCIASCHIKHOF}. We now ask for ultrametric version of Bourgain-Figiel-Milman theorem. \begin{problem} (\textbf{Ultrametric Bourgain-Figiel-Milman Problem}) $\mathbb{K}$ be a non-Archimedean valued field. For every $\varepsilon >0$, whether there is a universal constant $C(\varepsilon, \mathbb{K})>0$ satisfying the following: Every finite ultrametric space $\mathcal{M}$ contains a subset $\mathcal{N}$ satisfying following conditions. \begin{enumerate}[\upshape(i)] \item There is an injective map $f: \mathcal{N} \to c_0(\mathbb{N}, \mathbb{K})$ such that \begin{align*} d_{BFM}(\mathcal{N}, f(\mathcal{N}))\leq 1+\varepsilon. \end{align*} \item \begin{align*} o(\mathcal{N})\geq C(\varepsilon, \mathcal{A})\log (o(\mathcal{M})). \end{align*} \end{enumerate} \end{problem} \section{Ultrametric Enflo Type and Mendel-Naor Cotype Problems} Let $\mathcal{H}$ be a Hilbert space, $n\in \mathbb{N}$. Recall that for any $n$ points $h_1, \dots , h_n \in \mathcal{H}$, we have \begin{align}\label{H2} \frac{1}{2^n}\sum_{\varepsilon_1, \dots, \varepsilon_n \in \{-1, 1\}} \left\|\sum_{j=1}^{n}\varepsilon_j h_j\right\|^2=\sum_{j=1}^{n}\|h_j\|^2. \end{align} Equality (\ref{H2}) motivated the definition of Type and Cotype for Banach spaces. \begin{definition}\cite{ALBIACKALTON} Let $1\leq p \leq 2$. A Banach space $\mathcal{X}$ is said to be of \textbf{(Rademacher) Type $p$} if there exists $T_p(\mathcal{X})>0$ such that \begin{align*} \left(\frac{1}{2^n}\sum_{\varepsilon_1, \dots, \varepsilon_n \in \{-1, 1\}} \left\|\sum_{j=1}^{n}\varepsilon_j x_j\right\|^p\right)^\frac{1}{p} \leq T_p(\mathcal{X})\left(\sum_{j=1}^{n}\|x_j\|^p\right)^\frac{1}{p}, \quad \forall x_1, \dots , x_n \in \mathcal{X}, ~\forall n \in \mathbb{N}. \end{align*} \end{definition} \begin{definition}\cite{ALBIACKALTON} Let $2\leq q <\infty$. A Banach space $\mathcal{X}$ is said to be of \textbf{(Rademacher) Cotype $q$} if there exists $C_q(\mathcal{X})>0$ such that \begin{align*} \left(\sum_{j=1}^{n}\|x_j\|^q\right)^\frac{1}{q}\leq C_q(\mathcal{X}) \left(\frac{1}{2^n}\sum_{\varepsilon_1, \dots, \varepsilon_n \in \{-1, 1\}} \left\|\sum_{j=1}^{n}\varepsilon_j x_j\right\|^q\right)^\frac{1}{q}, \quad \forall x_1, \dots , x_n \in \mathcal{X}, ~\forall n \in \mathbb{N}. \end{align*} \end{definition} There is a vast literature on Type-Cotype of Banach spaces, see \cite{LATALAOLESZKIEWICZ, SZAREK, HANDBOOK1, HANDBOOK2, ALBIACKALTON, LIQUEFFELEC1, LIQUEFFELEC2, DIESTELJARCHOWTONGE, IVANISVILIVANHANDELVOLBERG, MAUREY}. In 1970, Enflo introduced the notion of Type for metric spaces \cite{ENFLO, MENDELNAOR}. \begin{definition} \cite{ENFLO, MENDELNAOR} Let $p>0$. A metric space $(\mathcal{M}, d)$ is said to be of \textbf{Enflo Type} $p$ if there exists $E_{p}(\mathcal{M})>0$ satisfying following conditions: For every $n \in \mathbb{N}$ and for every $f: \{-1, 1\}^n \to \mathcal{M}$ it holds \begin{align*} &\left(\frac{1}{2^n}\sum_{\varepsilon_1, \dots, \varepsilon_n \in \{-1, 1\}} d(f(\varepsilon_1, \dots, \varepsilon_n), d(-\varepsilon_1, \dots, -\varepsilon_n))^p\right)^\frac{1}{p} \leq\\ & E_p(\mathcal{M})\left(\sum_{j=1}^{n}\frac{1}{2^n}\sum_{\varepsilon_1, \dots, \varepsilon_n \in \{-1, 1\}} d(f(\varepsilon_1, \dots, \varepsilon_{j-1}, \varepsilon_{j}, \varepsilon_{j+1}, \dots, \varepsilon_n), d(\varepsilon_1, \dots, \varepsilon_{j-1}, -\varepsilon_{j}, \varepsilon_{j+1}, \dots, \varepsilon_n))^p\right)^\frac{1}{p}. \end{align*} \end{definition} In 2008, Mendel and Naor introduced the notion of Cotype for metric spaces \cite{MENDELNAOR}. \begin{definition} \cite{MENDELNAOR} Let $q>0$. A metric space $(\mathcal{M}, d)$ is said to be of \textbf{Mendel-Naor Cotype} $q$ if there exists $\Gamma_{q}(\mathcal{M})>0$ satisfying following conditions: For every $n \in \mathbb{N}$, there exists an even integer $m \in \mathbb{N}$ such that for every $f: \mathbb{Z}_m^n \to \mathcal{M}$ it holds \begin{align*} &\sum_{j=1}^n\frac{1}{m^n}\sum_{x \in \mathbb{Z}_m^n}d\left(f\left(x+\frac{m}{2}e_j, f(x)\right)\right)^q \\ &\leq \Gamma_{q}(\mathcal{M})^qm^q\frac{1}{m^n}\sum_{x \in \mathbb{Z}_m^n}\frac{1}{3^n} \sum_{\varepsilon \in \{-1, 0, 1\}^n}d(f(x+\varepsilon), f(x))^q, \quad \varepsilon=(\varepsilon_1, \dots, \varepsilon_n), \end{align*} where $\{e_j\}_{j=1}^n$ is the standard basis for $\mathbb{R}^n$. \end{definition} We now ask for ultrametric versions of Enflo Type and Mendel-Naor Cotype. \begin{problem} \textbf{(Ultrametric Enflo Type Problem)} Whether there is a way to define ultrametric Enflo Type? \end{problem} \begin{problem} \textbf{(Ultrametric Mendel-Naor Cotype Problem)} Whether there is a way to define ultrametric Mendel-Naor Cotype? \end{problem} Note that there are also notions of Type for metric spaces by Bourgain, Milman and Wolfson \cite{BOURGAINMILMANWOLFSON} and by Ball \cite{BALL}. 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