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Let $\mathbb B^n$ be a unit ball in $\mathbb R^n$ and let $F$ be a smooth function on it. Let $\mathbb B^k$ be a unit ball in $\mathbb R^k$.

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

1. For any point $x$ in the interior of $\mathbb B^k$ the minimum of $F|_{\varphi^{-1}(x)}$ is attained in the interior of $\mathbb B^n$.

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

3. Moreover, the strict minimum of $F$ on $\varphi^{-1}(\bf y)$ lies in the interior of $\mathbb B^n$.

Comment. I hope that the answer is positive if the differential of $\varphi$ is surjective on the whole ball $\mathbb B^n$ (but the comments of Fedja were producing counter-examples to the previous versions of the question).

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

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...).

Let $\mathbb B^n$ be a unit ball in $\mathbb R^n$ and let $F$ be a smooth function on it. Let $\mathbb B^k$ be a unit ball in $\mathbb R^k$.

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

1. For any point $x$ in the interior of $\mathbb B^k$ the minimum of $F|_{\varphi^{-1}(x)}$ is attained in the interior of $\mathbb B^n$.

2. The maximum $$\max_{y\in B}\min_{x\in \varphi^{-1}(y)} F(x)$$ is attained in the interior of $\mathbb B^k$.

Comment. The answer is clearly 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 which sounds less likely...).

Let $\mathbb B^n$ be a unit ball in $\mathbb R^n$ and let $F$ be a smooth function on it. Let $\mathbb B^k$ be a unit ball in $\mathbb R^k$.

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

1. For any point $x$ in the interior of $\mathbb B^k$ the minimum of $F|_{\varphi^{-1}(x)}$ is attained in the interior of $\mathbb B^n$.

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

3. Moreover, the strict minimum of $F$ on $\varphi^{-1}(\bf y)$ lies in the interior of $\mathbb B^n$.

Comment. I hope that the answer is positive if the differential of $\varphi$ is surjective on the whole ball $\mathbb B^n$ (but the comments of Fedja were producing counter-examples to the previous versions of the question).

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

Let $\mathbb B^n$ be a unit ball in $\mathbb R^n$ and let $F$ be a smooth function on it. Let $\mathbb B^k$ be a unit ball in $\mathbb R^k$.

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

1. For any point $x\in \mathbb B^k$ the minimum of $F|_{\varphi^{-1}(x)}$ is attained in the interior of $\mathbb B^n$.

2. The maximum $$\max_{y\in B}\min_{x\in \varphi^{-1}(y)} F(x)$$ is attained in the interior of $\mathbb B^k$.

Comment. The answer is clearly 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 which sounds less likely...).

Let $\mathbb B^n$ be a unit ball in $\mathbb R^n$ and let $F$ be a smooth function on it. Let $\mathbb B^k$ be a unit ball in $\mathbb R^k$.

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

1. For any point $x$ in the interior of $\mathbb B^k$ the minimum of $F|_{\varphi^{-1}(x)}$ is attained in the interior of $\mathbb B^n$.

2. The maximum $$\max_{y\in B}\min_{x\in \varphi^{-1}(y)} F(x)$$ is attained in the interior of $\mathbb B^k$.

Comment. The answer is clearly 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 which sounds less likely...).