Let $f: \mathbb{R}^N \to \mathbb{R}$ be a $\alpha$-strongly convex and $\beta$-strongly smooth function, i.e., $$ f(x) + \langle\nabla f(x), y- x\rangle + \frac{\alpha}{2}\|y-x\|^2 \leq f(y) \leq f(x) + \langle\nabla f(x), y- x\rangle + \frac{\beta}{2}\|y-x\|^2$$ where the norm is Euclidean norm. Let $x^*$ be the unique global optimum of $f$.
Now let $x_1, y_1, x_2, y_2$ be such that $f(x_1) = f(x_2) < f(y_1) = f(y_2)$, and $y_1 - x^* = k_1(x_1 - x^*)$ and $y_2 = k(x_2 - x^*)$ for some real numbers $k_1$ and $k_2$. Diagram to help you understand the setup.
Then, $\|x_1 - x^*\| / \|x_2 - x^*\|$ and $\|y_1 - x^*\| / \|y_2 - x^*\|$ are within the factor of $\beta / \alpha$. It is easy to show this because for any two points $x_1, x_2$ such that $f(x_1) = f(x_2)$, we have that $$\sqrt{\frac{\alpha}{\beta}} \leq \frac{\|x_1 - x^*\|}{\|x_2 - x^*\|} \leq \sqrt{\frac{\beta}{\alpha}}.$$
Now I want to relate $\|y_1 - x_1\| / \|y_1 - x^*\|$ and $\|y_2 - x_2\| / \|y_2 - x^*\|$. For the simple case of ellipsoids, this is also within factor of $\beta / \alpha$, and I believe that it should generalize to all strongly convex smooth functions. However, I cannot prove it or find the result in the literature. Does anyone know the proof for this?
