It seems that when optimising some function constrained to a submanifold (with corners) of its domain there is a higher probability that the optimum will be found in the singular part of the manifold (boundary or corners). In the GIF below the optimum really seems to spend more time on the corners than on the edges. My intuition is that when you get to a corner your degrees of freedom drop, so in the picture maybe the optimum wants to move in a direction it cant anymore so it gets stuck there?? It seems that this more precise statement should be true:
Let $M\subset V$ be a stratified subspace of some space $V$, $F$ be some family of smooth functions $V \to \mathbf{R}$ such that for all $f\in F$, $\max(f) \cap M = \varnothing$. For some point $x\in M$ denote by $c(x)$ the codimension of the strata of $M$ containing $x$, ie if $x$ is in the interior $c(x)=0$, if $x$ is in the boundary $c(x)=1$. Denote by $x_{f,M}$ a max of $f$ constrained to $M$ and let $F_{d,M}$ be the set of functions such that they have a max in the $d$-dimensional strata of $M$. Let $p$ be some probability on $F$ and $x_f$ be a maximum of $f$, then it seems that if $d<d'$ are such that the $d$-dimensional and $d'$-dimensional strata of M are non-empty, then $$p(F_{d,M})\geq p(F_{d',M})$$
Or maybe something thats a bit more vague would be easier like $ \mathbf{E}_p(c(x_{f,M}))$ is close to the dimension of $M$? Anyone knows if this is true? I restrict to the case of functions that dont have a max in the interior of $M$ because suppose $p$ has support on functions that have a max in $M$ then the statement is obviously false.
