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I'm trying to prove a simple proposition wich is in Burago's "A Course in Metric Spaces" (Exercise $8.4.5$, p.$287$). Before exposing my problem, let me give some definitions.

A metric space $(X,d)$ is said to be geodesic is any two points $a,b\in X$ can be joined by a lipschitz curve $\gamma$ such that $d(a,b)=\ell(\gamma)$, where $\ell$ denotes the lenght of $\gamma$. We denote $[a,b]$ the image of such a curve (called geodesic) joining $a,b$. Also, given $a,b,p\in X$ we consider the number

$$(a,b)_c=\frac{1}{2}(d(a,c)+d(c,b)-d(a,b)).$$

I'm trying to prove the following (which is true, I think...)

Suppose that $(X,d)$ is a geodesic metric space. Let $a,b,c\in X$ and consider the geodesic triangle whose vertices are $a,b,c$. Suppose that for all $p\in[b,c]$ we have

$$(a,b)_p\leqslant\delta.$$

Then,

$$d(p,[a,b])\leqslant\delta.$$

(this is not exactly the exercise, but, I think this is true. If it were, I think I can prove that any geodesic metric space satisfying some likely condition is $\delta$-hyperbolic)

This intuitively says, I think, that if the triangle $a,b,p$ is "thin", then the point $p$ is inside the $\delta$ neighborhood of $[a,b]$. I have tried lots of estimatives using the triangular inequality, in so many ways, that if this is true, this may be that kind of thing where we don't see the obvious... I couldn't figure out the argument. The signs of things in my estimatives seem to be all wrong.

If anyone could help, It would be really appreciated. Thanks again!

matgaio
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1 Answers1

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I was wrong, indeed. The true statements are:

1) If $(X,d)$ is $\delta$-hyperbolic, $\Delta abc$ is a geodesic triangle, then

$$(a,b)_p\leqslant d(p,[a,b])\leqslant(a,b)_p+2\delta.$$

The proof of this fact is written in the book. The exercise I was trying to figure out how to prove was:

2) Suppose that for all triangles $\Delta abc$ and for all points $p\in[b,c]$ it holds that $$\min((a,b)_p,(a,c)_p)\leqslant\delta$$ for an uniform $\delta.$ Then $(X,d)$ is $3\delta$-hyperbolic.

Proof: We have to show that the side $[b,c]$ is contained in the $3\delta$-neighborhood of the union $[a,b]\cup[a,c]$. Let $p\in[b.c]$. Then by hypothesis, we have that $$(a,b)_p\leqslant\delta\phantom{20}\textrm{or}\phantom{20}(a,c)_p\leqslant\delta.$$ If, for example, $(a,b)_p\leqslant\delta$, then by the preceeding result we have immediately \begin{eqnarray} d(p,[a,b])&\leqslant&(a,b)_p+2\delta\\ &\leqslant&\delta+2\delta\\ &=&3\delta \end{eqnarray} The same argument if it holds that $(a,c)_p\leqslant\delta$. Then we have that $[b,c]$ is contained in the $3\delta$-neighborhood of $[a,b]\cup[a,c]$. Therefore, $(X,d)$ is $3\delta$-hyperbolic.

matgaio
  • 2,273