convexity and center of mass in riemannian manifolds every riemannian manifold M is locally convex. Let $U$ be an open convex subset of $M$. Let $x_0, \ldots, x_n$ be points in $U$. Consider the map $\sigma \: \Delta \to U$ (where $\Delta$ is the standard $n$-simplex, i.e. the convex hull of the vectors $0 = e_0, e_1, \ldots, e_n$ in $\mathbb{R}^n$) such that $\sigma(e_i) = x_i$ for $0 \le i \le n$, extended over the whole $\Delta$ using "barycentric coordinates" (see "center of mass" on the reference below):
$$ \sigma(\lambda_1 e_1 + \cdots + \lambda_n e_n) = \sum_{i=0}^n \lambda_i x_i $$
(where $\sum_i \lambda_i \le 1$, $0 \le \lambda_i$).
Let $\tau \: I \to \Delta$ a constant speed segment. Is $\sigma \circ \tau$ a (constant speed) geodesic? In other words: is the simplex $\sigma$ geodesic?
See more at page 245 of berger: a panoramic view on riemannian geometry.
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[edit] I believe (hope) that this is true because it seems intuitively true, it would be nice, and there is a passage on Berger's book where, after he defines (what I call) the barycentric coordinates, saying:
" the resulting points for all possible weights fill out the \emph{convex closure} of the set {x_i}"

This is already false for surfaces. As an example, take any connected surface $(M,g)$ on nonconstant curvature. If your conjecture were true then every point $p\in M$ would admit a neighborhood $U$ and a geodesic mapping $f: U\to R^2$, i.e., a diffeomorphism to its image sending geodesics to geodesics. It was proven in Beltrami in 19th century that existence of such mapping forces $g$ to have constant curvature on $U$. See Theorem 95.1 in "Differential Geometry", by Erwin Kreyszig. 
