Perhaps this is basic knowledge in Riemannian geometry, but I can't seem to figure out the answer. Here is the precise statement of my question.

Let $M$ be a Riemannian manifold, $p$ a point in $M$. Let $R$ be small enough that $exp_p$ restricts to a diffeomorphism on the ball $B_R(0)$ of radius $R$ centered at the origin, and let $U_R$ be the intersection of $B_R(0)$ and any two dimensional plane through the origin in $T_p M$. Question: does there exist $R$ such that $exp_p(U_R)$ is geodesically convex, in the sense that for every two points of $exp_p(U_R)$ the unique geodesic segment connecting them lies entirely in $exp_p(U_R)$?

It would be really convenient for me if the answer is yes. If so, I am curious to know if the statement is still true if $P$ is replaced by a subspace of higher dimension, but I only need the result for planes.

Thanks!