A map with a small bound on the ratio of Dini derivatives is injective on the unit ball in Hilbert spaces

Question

Let $f$ be a continuous map defined from the unit ball $U$ of a Hilbert space $E$ to another Hilbert space $F$.

The Dini derivatives are defined as (for $x\in U$) $$D^+f(x)=\limsup_{y\to x}\frac{\Vert f(y)-f(x)\Vert}{\Vert y-x\Vert}$$ and $$D^-f(x)=\liminf_{y\to x}\frac{\Vert f(y)-f(x)\Vert}{\Vert y-x\Vert}$$

I have read that if on $U$ we have $0<m\leq D^-f(x)\leq D^+f(x)\leq M<\infty$ and $$k=M/m<\sqrt{(1+\sqrt{5})/2}$$ then one can prove that $f$ is a homeomorphism from $U$ to $f(U)$

More precisely, for $x, y\in U$ we have $$\Vert f(x)-f(y)\Vert\geq \mu\Vert x-y\Vert$$ where $$\mu=m\frac{(1+k\sqrt{k^2-1})}{1+\sqrt{k^2-1}}$$ Does anybody know a proof of this strange result or a reference for a proof of it (that seems to work only because of the geometry of Hilbert space, not in a more general Banach setting)?

Answer 1

This was originally proved by Fritz John in [281] (references at the end). John called a map with the conditions you stated "$(M, m)$-quasi-isometric" but I would not use this term for this concept today, when "quasi-isometry" is associated more with global two-point conditions than with infinitesimal derivative-like bounds.

The result was improved by Julian Gevirtz [202] in several ways:

• The assumption can be weakened to $M/m < \sqrt{1+\sqrt{2}}$
• One can replace $F$ with an arbitrary Banach space, provided $M/m < \sqrt{2}$
• One can replace both $E$ and $F$ with arbitrary Banach spaces, provided $M/m < \mu_0 = 1.114\dots $ where $\mu_0$ is a solution of some algebraic equation.

In book form, these results appear in Chapter 14 of "Geometric Nonlinear Functional Analysis" by Yoav Benyamini and Joram Lindenstrauss. I quote from the notes at the end of this chapter, pp. 356-357:

The theory of quasi-isometric maps was initiated by F. John in [281]. Motivated by problems in elasticity, he embarked in [281] and later in [282]-[284] on a purely geometric study. Most of the results in this chapter are due to him.

The main injectivity result (Theorem 14.10) was proved by John only in Hilbert space, and he used special properties of Hilbert space geometry. The generalization to Banach spaces is due to J. Gevirtz [202], and we presented here his proof.

There are injectivity results for functions defined on more general sets. A domain $\Omega$ is called a uniform domain if there is a constant $b>0$ so that for each $u,v \in\Omega$ there is a path $\phi(t)$ [of length $L$] connecting $u$ and $v$ and parametrizaed by arc length so that $B(\phi(t), b\min(t, L-t))\subset \Omega$ for all $t$. Gevirtz [201] (improving results and estimates of John in the Hilbert space setting) proved that there is a constant $\mu(b)>0$, independed of the Banach space, such that if $f$ is a quasi-isometry on a uniform domain with constant $b$, and if $M/m < \mu(b)$, then $f$ is injective.

References

[201] J. Gevirtz, Injectivity in Banach spaces and the Mazur-Ulam theorem on isometries. Trans. Amer. Math. Soc. 274 (1982), no. 1, 307–318.

[202] J. Gevirtz, Injectivity of quasi-isometric mappings of balls. Proc. Amer. Math. Soc. 85 (1982), no. 3, 345–349. Link to article

[281] F. John, Quasi-isometric mappings in Hilbert space, New York Univ., Courant Inst. Math. Sci., Res. Rep. No. IMM-NYU 336, 1965.

[282] F. John, On quasi-isometric mappings, I, Comm. Pure Appl. Math. 21 (1968), 77-110.

[283] F. John, On quasi-isometric mappings, II, Comm. Pure Appl. Math. 22 (1969), 265-278.

[284] F. John, Note on the paper "On quasi-isometric mappings. I''. Comm. Pure Appl. Math. 25 (1972), 497.