Lipschitz Condition and Optimization

Question

I am reading paper on optimization. I am trying to understand the following statement:

I would like to ask how the Ball doesnot contain the minimum of f? I would appreciate if this could be graphically shown something like following taken from Bayesian Optimization and Data Science

Answer 1

Let $x^* \in \mathcal{X}$ be the point at which the function attains the minimum value. Then by definition, we have $f(x^*) = M$. Let us apply the Lipschitz property between $x_j$ and $x^*$. Thus, we have $|f(x_j) - f(x^*)| \leq L \| x_j - x^*\| \implies \| x_j - x^*\| \geq \dfrac{f(x_j) - f(x^*)}{L} = r_j $. Thus, the point $x^* \notin B_{r_j}(x_j)$. More specifically, it does not lie in the interior of the ball.