On possible counterexample.

Let $M$ be a Riemannian manifold and $x,y,z\in M$. One can measure the maximal radus of ball inside of convex hull of $\{x,y,z\}$, let it be $r(M,x,y,z)$.

Is it possible to find a a sequence $M_n$ of negatively curved $n$-dimensional manifolds with points $x_n,y_n,z_n\in M_n$ such that $|x_ny_n|=|y_nz_n|=|z_nx_n|=1$ and $r(M_n,x_n,y_n,z_n)$ stays bounded away from zero as $n\to\infty$?

On a possible counterexample. Let $M$ be a Riemannian manifold and $x,y,z\in M$. One can measure the maximal radius of a ball inside of the convex hull of $\{x,y,z\}$, let it be $r(M,x,y,z)$.

Is it possible to find a a sequence $M_n$ of negatively curved $n$-dimensional manifolds with points $x_n,y_n,z_n\in M_n$ such that $|x_ny_n|=|y_nz_n|=|z_nx_n|=1$ and $r(M_n,x_n,y_n,z_n)$ stays bounded away from zero as $n\to\infty$?

If the answer is "yes", then it should lead to a counterexample. The argument in our "About every convex..." might help.

On possible counterexample.

Let $M$ be a Riemannian manifold and $x,y,z\in M$. One can measure the maximal radus of ball inside of convex hull of $\{x,y,z\}$, let it be $r(M,x,y,z)$.

Is it possible to find a a sequence $M_n$ of negatively curved $n$-dimensional manifolds with points $x_n,y_n,z_n\in M_n$ be three points on unit distance such that $r(M_n,x_n,y_n,z_n)$ stays bounded away from zero as $n\to\infty$?

On possible counterexample.

Let $M$ be a Riemannian manifold and $x,y,z\in M$. One can measure the maximal radus of ball inside of convex hull of $\{x,y,z\}$, let it be $r(M,x,y,z)$.

Is it possible to find a a sequence $M_n$ of negatively curved $n$-dimensional manifolds with points $x_n,y_n,z_n\in M_n$ such that $|x_ny_n|=|y_nz_n|=|z_nx_n|=1$ and $r(M_n,x_n,y_n,z_n)$ stays bounded away from zero as $n\to\infty$?