WorksMathematics

Ultrametric Bourgain-Figiel-Milman, Enflo Type and Mendel-Naor Cotype Problems

Published Oct 31, 2026Fresh — published within the last 30 daysv13 min read

AI contributions

Not recorded for this paper

Abstract

We ask for ultrametric version of following three: (1) Bourgain-Figiel-Milman Theorem, (2) Enflo Type, (3) Mendel-Naor Cotype.

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 Banach-Mazur distance between $\mathcal{X}$ and $\mathcal{Y}$ is defined as

d_{BM}(\mathcal{X},\mathcal{Y}) := \inf\{\lVert T\rVert\lVert T^{-1}\rVert: T:\mathcal{X}\to\mathcal{Y}\text{ is invertible linear operator}\}.

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.

Theorem 1.1. [1, 6] (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

k \le C\frac{\varepsilon^{2}}{\lvert\log\varepsilon\rvert}\log n,

there exists a $k$ -dimensional Banach subspace $\mathcal{Y}$ of $\mathcal{X}$ such that

d_{BM}(\mathcal{Y},(\mathbb{R}^{k},\langle\cdot,\cdot\rangle)) < 1+\varepsilon.

We refer [8, 9, 10, 11, 12, 13, 22, 23, 26, 27, 28, 30, 31, 32, 34] for more information on Dvoretzky-Milman theorem. With Theorem 1.1, 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

\lVert f\rVert_{\mathrm{Lip}} := \sup\left\{\frac{d(f(x),f(y))}{d(x,y)}:x,y\in\mathcal{M},x\ne y\right\}.

The Bourgain-Figiel-Milman distance between $\mathcal{M}$ and $\mathcal{N}$ is defined as

d_{BFM}(\mathcal{M},\mathcal{N}) := \inf\{\lVert f\rVert_{\mathrm{Lip}}\lVert f^{-1}\rVert_{\mathrm{Lip}}:f:\mathcal{M}\to\mathcal{N}\text{ is injective}\}.

In 1986, Bourgain, Figiel and Milman proved the following nonlinear Dvoretzky theorem [3].

Theorem 1.2. [3] (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.

(i) There is an injective map $f : \mathcal{N} \to\ell^2(\mathbb{N},\mathbb{R})$ such that

d_{BFM}(\mathcal{N},f(\mathcal{N})) \leq1+\varepsilon.

(ii)

o(\mathcal{N}) \geq C(\varepsilon)\log(o(\mathcal{M})).

Further, following holds: If $\varepsilon< 1$ , then we can take

C(\varepsilon) = \frac{c_1\varepsilon}{\log\left(\frac{c_2}{\varepsilon}\right)}

for some $c_1 > 0,c_2 > 0$ .

We will formulate problem based on Theorem 1.2 to ultrametric spaces.

Definition 1.3. [33] Let $\mathcal{M}$ be a set. A map $d : \mathcal{M} \times\mathcal{M} \to[0,\infty)$ is said to be an ultrametric if it satisfies following conditions.

(i) $d(z,z)=0$ for all $z \in\mathcal{M}$ . If $x,y \in\mathcal{M}$ are such that $d(x,y)=0$ , then $x=y$ .

(ii) $d(x,y)=d(y,x)$ for all $x,y \in\mathcal{M}$ .

(iii) $d(x,y) \leq\max\{d(x,z),d(z,y)\}$ for all $x,y,z \in\mathcal{M}$ .

In this case, $(\mathcal{M},d)$ is called as an ultrametric space.

Let $\mathbb{K}$ be a non-Archimedean valued field. We define

c_0(\mathbb{N},\mathbb{K}) := \left\{\{a_n\}_n : a_n \in\mathbb{K},\ \forall n \in\mathbb{N},\ \lim_{n\to\infty} a_n=0\right\}

equipped with the ultra-norm

\left\|\{a_n\}_n\right\| := \max_{n\in\mathbb{N}} |a_n|,\quad\forall\{a_n\}_n \in c_0(\mathbb{N},\mathbb{K})

and the p-adic inner product

\langle\{a_n\}_n,\{b_n\}_n\rangle:= \sum_{n=1}^{\infty}a_nb_n,\quad\forall\{a_n\}_n,\{b_n\}_n \in c_0(\mathbb{N},\mathbb{K}).

Then $c_0(\mathbb{N},\mathbb{K})$ is a non-Archimedean Banach space which is also a p-adic Hilbert space [17, 29]. We now ask for ultrametric version of Bourgain-Figiel-Milman theorem.

Problem 1.4. (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.

(i) There is an injective map $f : \mathcal{N} \to c_0(\mathbb{N},\mathbb{K})$ such that

d_{BFM}(\mathcal{N},f(\mathcal{N})) \leq1+\varepsilon.

(ii)

o(\mathcal{N}) \geq C(\varepsilon,\mathbb{K})\log(o(\mathcal{M})).

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 $h_1,\ldots,h_n \in\mathcal{H}$ , we have

\frac{1}{2^n}\sum_{\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}}\left\|\sum_{j=1}^{n}\varepsilon_jh_j\right\|^2=\sum_{j=1}^{n}\|h_j\|^2. \tag*{(1)}

Equality (1) motivated the definition of Type and Cotype for Banach spaces.

Definition 2.1. [1] Let $1 \le p \le2$ . A Banach space $\mathcal{X}$ is said to be of (Rademacher) Type $p$ if there exists $T_p(\mathcal{X}) > 0$ such that

\left(\frac{1}{2^n}\sum_{\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}}\left\|\sum_{j=1}^{n}\varepsilon_jx_j\right\|^p\right)^{\frac{1}{p}}\le T_p(\mathcal{X})\left(\sum_{j=1}^{n}\|x_j\|^p\right)^{\frac{1}{p}},\qquad\forall x_1,\ldots,x_n\in\mathcal{X},\ \forall n\in\mathbb{N}.

Definition 2.2. [1] Let $2 \le q < \infty$ . A Banach space $\mathcal{X}$ is said to be of (Rademacher) Cotype $q$ if there exists $C_q(\mathcal{X}) > 0$ such that

\left(\sum_{j=1}^{n}\|x_j\|^q\right)^{\frac{1}{q}}\le C_q(\mathcal{X})\left(\frac{1}{2^n}\sum_{\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}}\left\|\sum_{j=1}^{n}\varepsilon_jx_j\right\|^q\right)^{\frac{1}{q}},\qquad\forall x_1,\ldots,x_n\in\mathcal{X},\ \forall n\in\mathbb{N}.

There is a vast literature on Type-Cotype of Banach spaces, see [1, 5, 14, 15, 16, 19, 20, 21, 24, 35].

In 1970, Enflo introduced the notion of Type for metric spaces [7, 25].

Definition 2.3. [7, 25] Let $p > 0$ . A metric space $(\mathcal{M},d)$ is said to be of 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

\left(\frac{1}{2^n}\sum_{\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}}d\bigl(f(\varepsilon_1,\ldots,\varepsilon_n),d(-\varepsilon_1,\ldots,-\varepsilon_n)\bigr)^p\right)^{\frac{1}{p}} \le E_p(\mathcal{M})\left(\sum_{j=1}^{n}\frac{1}{2^n}\sum_{\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}}d\bigl(f(\varepsilon_1,\ldots,\varepsilon_{j-1},\varepsilon_j,\varepsilon_{j+1},\ldots,\varepsilon_n),d(\varepsilon_1,\ldots,\varepsilon_{j-1},-\varepsilon_j,\varepsilon_{j+1},\ldots,\varepsilon_n)\bigr)^p\right)^{\frac{1}{p}}.

In 2008, Mendel and Naor introduced the notion of Cotype for metric spaces [25].

Definition 2.4. [25] Let $q > 0$ . A metric space $(\mathcal{M},d)$ is said to be of 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

\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\right),f(x)\right)^q \le\Gamma_q(\mathcal{M})^q m^q\frac{1}{m^n}\sum_{x\in\mathbb{Z}_m^n}\sum_{\varepsilon\in\{-1,0,1\}^n}d(f(x+\varepsilon),f(x))^q,\qquad\varepsilon=(\varepsilon_1,\ldots,\varepsilon_n),

where $\{e_j\}_{j=1}^n$ is the standard basis for $\mathbb{R}^n$ .

We now ask for ultrametric versions of Enflo Type and Mendel-Naor Cotype.

Problem 2.5. (Ultrametric Enflo Type Problem) Whether there is a way to define ultrametric Enflo Type?

Problem 2.6. (Ultrametric Mendel-Naor Cotype Problem) Whether there is a way to define ultrametric Mendel-Naor Cotype?

Note that there are also notions of Type for metric spaces by Bourgain, Milman and Wolfson [4] and by Ball [2]. These lead to following problems.

Problem 2.7. (Ultrametric Bourgain-Milman-Wolfson Type Problem) Whether there is a way to define ultrametric Bourgain-Milman-Wolfson Type?

Problem 2.8. (Ultrametric Markov/Ball Type Problem) Whether there is a way to define ultrametric Markov/Ball Type?

Remark 2.9. In 2026, we formulated non-Archimedean John, non-Archimedean Dvoretzky-Milman, non-Archimedean Type-Cotype and non-Archimedean Kwapien problems which are still open [18].

References

[1]Fernando Albiac and Nigel J. Kalton. Topics in Banach space theory, volume 233 of Graduate Texts in Mathematics. Springer, [Cham], 2016.
[2]K. Ball. Markov chains, Riesz transforms and Lipschitz maps. Geom. Funct. Anal., 2(2):137–172, 1992.DOI
[3]J. Bourgain, T. Figiel, and V. Milman. On Hilbertian subsets of finite metric spaces. Isr. J. Math., 55:147–152, 1986.DOI
[4]J. Bourgain, V. Milman, and H. Wolfson. On type of metric spaces. Trans. Am. Math. Soc., 294:295–317, 1986.DOI
[5]Joe Diestel, Hans Jarchow, and Andrew Tonge. Absolutely summing operators, volume 43 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995.
[6]Aryeh Dvoretzky. Some results on convex bodies and Banach spaces. In Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pages 123–160. Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961.
[7]Per Enflo. On the nonexistence of uniform homeomorphisms between Lp-spaces. Ark. Mat., 8:103–105, 1970.DOI
[8]T. Figiel. Some remarks on Dvoretzky’s theorem on almost spherical sections of convex bodies. Colloq. Math., 24:241–252, 1971/72.DOI
[9]T. Figiel. A short proof of Dvoretzky’s theorem. In Séminaire Maurey-Schwartz 1974–1975: Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. No. XXIII, pages 6 pp. (erratum, p. 3). 1975.
[10]T. Figiel. A short proof of Dvoretzky’s theorem on almost spherical sections of convex bodies. Compositio Math., 33(3):297–301, 1976.
[11]T. Figiel, J. Lindenstrauss, and V. D. Milman. The dimension of almost spherical sections of convex bodies. Acta Math., 139(1-2):53–94, 1977.
[12]Y. Gordon, O. Guedon, and M. Meyer. An isomorphic Dvoretzky’s theorem for convex bodies. Studia Math., 127(2):191–200, 1998.DOI
[13]Yehoram Gordon. Gaussian processes and almost spherical sections of convex bodies. Ann. Probab., 16(1):180–188, 1988.DOI
[14]Paata Ivanisvili, Ramon van Handel, and Alexander Volberg. Rademacher type and Enflo type coincide. Ann. Math. (2), 192(2):665–678, 2020.DOI
[15]W. B. Johnson and J. Lindenstrauss, editors. Handbook of the geometry of Banach spaces. Vol. 1. North-Holland Publishing Co., Amsterdam, 2001.
[16]W. B. Johnson and J. Lindenstrauss, editors. Handbook of the geometry of Banach spaces. Vol. 2. North-Holland, Amsterdam, 2003.
[17]G. K. Kalisch. On p-adic Hilbert spaces. Ann. Math. (2), 48:180–192, 1947.DOI
[18]K. Mahesh Krishna. Non-Archimedean John, non-Archimedean Dvoretzky-Milman, non-Archimedean type-cotype and non-Archimedean Kwapien problems. hexagon:2610.00653v1, 2026.
[19]Rafal Latala and Krzysztof Oleszkiewicz. On the best constant in the Khinchin-Kahane inequality. Studia Math., 109(1):101–104, 1994.
[20]Daniel Li and Herve Queffelec. Introduction to Banach spaces: analysis and probability. Vol. 1, volume 166 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2018.
[21]Daniel Li and Herve Queffelec. Introduction to Banach spaces: analysis and probability. Vol. 2, volume 167 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2018.
[22]J. Matousek. Lectures on discrete geometry, volume 212 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2002.
[23]Ivan Matsak and Anatolij Plichko. Dvoretzky’s theorem by Gaussian method. In Functional analysis and its applications, volume 197 of North-Holland Math. Stud., pages 171–184. Elsevier Sci. B. V., Amsterdam, 2004.DOI
[24]Bernard Maurey. Type, cotype and K-convexity. In Handbook of the geometry of Banach spaces. Volume 2, pages 1299–1332. Amsterdam: North-Holland, 2003.DOI
[25]Manor Mendel and Assaf Naor. Metric cotype. Ann. Math. (2), 168(1):247–298, 2008.
[26]V. Milman. Dvoretzky’s theorem—thirty years later. Geom. Funct. Anal., 2(4):455–479, 1992.DOI
[27]V. D. Milman. A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies. Funkcional. Anal. i Priložen., 5(4):28–37, 1971.
[28]Vitali D. Milman and Gideon Schechtman. Asymptotic theory of finite-dimensional normed spaces, volume 1200 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986.
[29]C. Perez-Garcia and W. H. Schikhof. Locally convex spaces over non-Archimedean valued fields, volume 119 of Camb. Stud. Adv. Math. Cambridge: Cambridge University Press, 2010.
[30]Gilles Pisier. Probabilistic methods in the geometry of Banach spaces. In Probability and analysis (Varenna, 1985), volume 1206 of Lecture Notes in Math., pages 167–241. Springer, Berlin, 1986.DOI
[31]Gilles Pisier. The volume of convex bodies and Banach space geometry, volume 94 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1989.DOI
[32]Gideon Schechtman. A remark concerning the dependence on ε in Dvoretzky’s theorem. In Geometric aspects of functional analysis (1987–88), volume 1376 of Lecture Notes in Math., pages 274–277. Springer, Berlin, 1989.
[33]W. H. Schikhof. Ultrametric calculus. An introduction to p-adic analysis, volume 4 of Camb. Stud. Adv. Math. Cambridge: Cambridge University Press, 2006.
[34]Andrzej Szankowski. On Dvoretzky’s theorem on almost spherical sections of convex bodies. Israel J. Math., 17:325–338, 1974.DOI
[35]S. J. Szarek. On the best constants in the Khinchin inequality. Studia Math., 58(2):197–208, 1976.DOI

Ultrametric Bourgain-Figiel-Milman, Enflo Type and Mendel-Naor Cotype Problems

Abstract

Ultrametric Bourgain-Figiel-Milman Problem

Ultrametric Enflo Type and Mendel-Naor Cotype Problems

References

References

Publication

Taxonomy

Authors

AI contributions

More in Mathematics

More in Mathematics