Let $\mathbb B^n$ be an open unit ball in $\mathbb R^n$ and let $F$ be a smooth function on it. Let $\frac{1}{2}\mathbb B^n\subset \mathbb B^n$ be an open ball or radius $\frac{1}{2}$. Let $\mathbb B^k$ be an open unit ball in $\mathbb R^k$, and $\frac{1}{2}\mathbb B^n\subset \mathbb B^n$ be an open ball or radius $\frac{1}{2}$.

**Question.** Is it true that $F$ has a critical point in $\mathbb B^n$ if there exists a smooth surjective map $\varphi:\mathbb B^n\to \mathbb B^k$, that has the following property:

1) For any $y\in \frac{1}{2}\mathbb B^k $ the *strict minimum*
$$\min_{x\in \varphi^{-1}(y)} F(x)$$
is attained in $\mathbb B^n$ at some point ${\bf x}\in \frac{1}{2}\mathbb B^n$. In particular, $\inf F$ on $(\varphi^{-1}(y)\cap (\mathbb B^n\setminus \frac{1}{2}\mathbb B^n))$ is smaller than $\min_{x\in \varphi^{-1}(y)} F(x)=F({\bf x})$ (if such an intersection is non-empty).

2) The point ${\bf y}\in \mathbb B_k$ where the maximum of $$\max_{y\in \mathbb B^k}\min_{x\in \varphi^{-1}(y)} F(x)$$ is attained lies in $\frac{1}{2}\mathbb B^k.$

**Comment.** This is a largely rewritten formulation. I hope that the answer is positive if the differential of $\varphi$ is surjective on the whole ball $\mathbb B^n$

I would be grateful for a reference (or a counter-example...).