## On Lipschitz ball noncollapsing functions and uniform co-Lipschitz mappings of the plane.(English)Zbl 1108.54019

A map $$f$$ between two metric spaces is co-Lipschitz, with constant $$C$$, if the image of a ball with centre $$x$$ and radius $$r$$ always contains the ball with centre $$f(x)$$ and radius $$Cr$$. Reversing this inclusion obviously yields the definition of a Lipschitz map. If we assume only that the image of a ball of radius $$r$$ always contains a ball with radius $$Cr$$, then $$f$$ is called ball noncollapsing.
The author had previously shown [Geometric aspects of functional analysis. Proceedings of the Israel seminar (GAFA) 2001–2002. Berlin: Springer. Lect. Notes Math. 1807, 148-157 (2003; Zbl 1058.46053)] that if $$f$$ is an $$L$$-Lipschitz and $$C$$-co-Lipschitz map from the plane to itself, then for every point $$x$$, the cardinality of $$f^{-1}(x)$$ is at most $$L/C$$. Here she reaches the same conclusion, under the weaker assumption that $$f$$ is uniformly continuous and $$C$$-co-Lipschitz. This time $$L$$ is replaced by the weak Lipschitz constant of $$f$$, which exists because uniformly continuous maps between normed spaces are Lipschitz for large distances. The $$n$$th power function on the complex plane shows that this estimate is sharp. Another interesting result is that any Lipschitz and co-Lipschitz map from the plane to itself does not collapse areas of measurable sets. In the previous paper, it had also been shown that if $$f$$ is an $$L$$-Lipschitz, $$C$$-ball noncollapsing map of the line (or the plane) to itself with $$L<2C$$, then $$f$$ is one-one; the absolute value function on the line shows that this estimate is sharp. In a real tour de force it is proved here that there is an $$L$$-Lipschitz, $$C$$-ball noncollapsing map $$f$$ of the line to itself and a point $$x$$ such that $$f^{-1}(x)$$ is infinite if, and only if, $$L>3C$$.

### MSC:

 54E40 Special maps on metric spaces 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 46T99 Nonlinear functional analysis

Lipschitz; plane

Zbl 1058.46053
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