Here is tricky problem I came across when writing a paper. But I can't figure it out, so I ask for help here.

Let $M$ be a $n$-dimensional smooth manifold which is also a complete geodesic space. We can map a small ball around a point in $M$ to $\mathbb{R}^n$ endowed with the Euclidean metric such that the map is a homeomorphism.

The small ball is a convex set in $M$ (i.e the geodesic lines which connect any two points in the small ball are still in it). Can we find a homeomorphism such that the image of the small ball in $\mathbb{R}^n$ is also convex set?