# Area of triangles vs. comparison triangles.

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