Distortion and Norm Stabilization

Question

I'm curious about the Distortion Problem in Banach space theory and its relation with norm stabilization. I found that if $(X, \| \cdot \|)$ is an infinite-dimensional separable Banach space, then $X$ does not contain a distortable subspace iff every equivalent norm on $X$ stabilizes. It seems to me that the proof of this equivalence is sort of "easy" for specialists. I've tried to come up with it, but so far I've failed every time; my guess is that one needs some kind of well-known trick to go, for example, from the fact that an equivalent norm $\lvert \cdot \rvert$ distorts $X$ to show that $\lvert \cdot \rvert : S_X \to \mathbb{R}$ does not stabilize.

Could someone provide a little hint on how to go about this?

Here you go the corresponding definitions:

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It may be useful if you provide the relevant definitions for "distortable" and "stabilizes". — Christopher A. Wong, Jul 17 '14 at 23:24
@ChristopherA.Wong: We say that $X$ is distortable if it is $\lambda$-distortable for some $\lambda > 1$. — ragrigg, Jul 17 '14 at 23:34

score 1 · Accepted Answer · answered Jul 17 '14 at 23:56

You want to show that there is a distortable subspace if and only if there is an equivalent norm that is not oscillation stable. You can make the connection between the two concepts the following: if $Y$ is a distortable subspace, then for an equivalent norm $|\cdot |$, $$ |y_1| / |y_2| \geq 1 + \delta$$ for some $\delta > 0$, and for $y_1, y_2$ on the unit sphere. But then

$$ |y_1| - |y_2| \geq |y_2| \delta \geq C \|y_2\| \delta = C \delta $$ for some $C > 0$. Therefore the equivalent norm $| \cdot |$ cannot be oscillation stable.

For the reverse direction, start with a non-stabilizing equivalent norm. Choose a subspace where the oscillation is therefore bounded from below, and show that this forms a distortable subspace using the same symbolic manipulations as above.

Nice answer! The "trick" I did not see was your first inequality; everything follows from it. I think that it is important to note that when you prove that the existence of a distortable subspace $Y$ gives you a non-stabilizing equivalent norm $\lvert \cdot \rvert$, we are implicity using the fact that we can extend $\lvert \cdot \rvert$ from $Y$ to an equivalent norm on $X$ (since, by definition, we need this norm to be defined on the whole space, not just on $Y$). — ragrigg, Jul 18 '14 at 00:32

Distortion and Norm Stabilization

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