For a standard unconstrained minimization problem:
$$\min_{x \in \mathcal{X}} f(x)$$
I am interested in understanding how "big" the set of $\epsilon$-optimal solutions "usually" are. Mathematically, consider the space $C[0,1]$, all continuous functions over $[0,1]$. Then I want to understand the "usual" behavior of $p(\epsilon)$ defined below: $$p(\epsilon) = |x \in [0,1] \mid f(x)-f^* \leq \epsilon|$$ where $f^*$ is the optimal minimum. How does this function grow as $\epsilon$ increases? Of course, one can try to solve $p(\epsilon)$ for special parametric classes of functions but I want to understand if there is some general understanding of $p(\epsilon)$ over all continuous functions (e.g. with any reasonable/natural probability distribution over these functions). I have tried to find any resources related to this problem online but I think I am probably searching the wrong keywords. Any pointers toward relevant literature is very helpful!