The setting is as follows:
Suppose we're on a complete Riemannian manifold $M$. Let $p\in M$ and $e_1\in T_p M$ (unit vector). Complete $e_1$ into an orthonormal basis $\{ e_1,\,\dots,\,e_n\}$ such that $$R(e_i,\,e_1)e_1 = k_i e_1$$ where $R$ denotes the curvature tensor and $k_i = \sec(e_1,\,e_i)$. Consider a large normal ball around $p$ and take $δ,\,r>0$ so that it includes $A_0 := B_δ(\exp_p (-re_1))$ and $A_1 := B_δ(\exp_p (re_1))$. Define $n$ numbers: $$δ_i:=δ \bigg[1+\frac{r^2}{2}\Big(k_i+\frac{ε}{2n}\Big)\bigg], \ i=1,\,\dots,\,n$$ and choose small enough $r$ and $δ$ so that the ellipsoid $$A_{1/2}:= \exp_p\bigg(\bigg\{v\in T_p M: \sum_{i=1}^n \bigg(\frac{v_i}{δ_i}\bigg)^2 \leq 1\bigg\}\bigg)$$ will also be included in the large normal ball.
What I wish to show is that for every geodesic $γ$ with $γ(0)\in A_0$ and $γ(1)\in A_1$ one must have $γ(1/2)\in A_{1/2}$. I don't really have a clue how to even start. What I'm thinking is that if we parallel translate the given basis along a geodesics joining the centers of $A_0$ and $A_1$ and Taylor expand the curve at $p$ we just want to show that its Jacobi field will be an element of the set inside the exponential map in the definition of $A_{1/2}$. Then, this Jacobi field's coefficients $(y_i)_{i=1}^n$ (with respect to the parallel basis) must satisfy an O.D.E. such as $$y_i''(t)+k_i y_i(t) + O(t) = 0.$$
I'm not even sure if my arguments are correct, since they are purely geometrical in nature and no actual math has been written down.
Any kind of help (hint, correction, solution) would be greatly appreciated!
Thank you in advance!