Let M be a Riemannian manifold in $\mathbb{R}^n$ of positive sectional curvature and S be a geodesically convex set on M and $T \in GL(n, \mathbb{R})$. Is it true that $T(S)$ is geodesically convex in $T(M)$? If not would it be true in special case of $M$ being hyperellipsoid?
Without curvature positivity you can cook up a counter-example to this statement: $M$ is flat everywhere except for a low ridge, $S$ covers that ridge. A geodesic in $S$ runs over this low ridge and remains in $S$. $T$ stretches the ridge upwards a lot and geodesic is forced to go around leaving $S$.
I also tried analyzing $||\ddot{\gamma}_{TM}(t)||$ via $||\ddot{\gamma}_{M}(t)||$ with respect to $\kappa (\partial TS)$ on the basis of Euler-Lagrange, with idea being that if boundary of $S$ starts off more curved than geodesic then perhaps after $T$ same relationship holds. The main difficulty is that E-L solution is not very friendly.