Here are some problems which I wasn't able to solve.


1. Let $\gamma(t) = (t,t^{2},f(t)) \in C^{3}(\mathbb{R})$ be a curve such that its torsion changes sign infinitely many times. The question is whether there exists finite algorithm which constructs minimal concave function $B(x_{1},x_{2})$ on the parabolic strip $\{ (x_{1},x_{2}) : x_{1}^{2}\leq x_{2} \leq x_{1}^{2}+1\}$ with the boundary condition $B(t,t^{2})=f(t)$

Comment: For some particular cases it is possible to present such algorithm. The question is whether it is true in such general case.

2. Let $p >1, \tau >0$. Try to find function $B(x_{1},x_{2},x_{3})$ defined on the set $\Omega = \{ (x_{1},x_{2},x_{3}) : |x_{1}|^{p} \leq x_{3}\}$ with the boundary condition $B(x_{1},x_{2},|x_{1}|^{p})=|x_{2}+\tau x_{1}|^{p}$ such that it is concave in the planes $x_{1}\pm x_{2}=C$ as $C$ ranges over the real line and it is minimal such zigzag concave function with the properties mentioned above

Cooment: Of course case $p=2$ is trivial.

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