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Let $X$ be a complete, simply connected Riemannian manifold satisfying a quadratic (coarse) isoperimetric inequality. (I.e., there is a constant $C_{0}$ such that every loop of length $\ell$ has a filling disk of area $\leq C_{0}\ell^{2}+C_{0}$.)

For points $a,b,c\in X$, define $Area_{X}(a,b,c)$ to be the minimal area of any geodesic triangle in $X$ with vertices $a,b,c$. Define $Area_{comp}(a,b,c)$ to be the area of a Euclidean triangle with side lengths $d(a,b)$, $d(b,c)$ and $d(c,a)$.

Does there exist a constant $C$ such that for all $a,b,c\in X$ we have:

$Area_{X}(a,b,c)\leq C Area_{comp}(a,b,c)+C$?

If the answer is no, then are there counterexamples when $X$ is homogeneous?

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No.

Blow a bubble of the same size at each integer point of $\mathbb R^2$. Clearly coarse isoperimetric inequality will hold.

On the other hand, the global metric on the plane can be made arbitrary close to the Manhattan metric, in particular there will be triangles which bound arbitrary large area while its comparison area is near zero.

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  • $\begingroup$ P.S. If you look in "Periodic metric" by Burago you will see more than you want to. $\endgroup$ – Anton Petrunin Feb 18 '13 at 0:17

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