On stability of a unit ball in Minkowski space with respect to self-area

The main results of the paper are the following two statements. If the length of the unit circle $\partial B=\{\|x\|=1\}$ on Minkowski plane $M^2$ is equal to $O(B)=8(1-\varepsilon)$, $0\le\varepsilon\le 0.04$, then there exists a parallelogram which is centrally symmetric with respect to the origin $o$ and the sides of which lie inside an annulus $(1+18\varepsilon)^{-1}\le\|x\|\le 1$. If the area of the unit sphere $\partial B$ in the Minkowski space $M^n$, $n\ge 3$, is equal to $O(B)=2n\cdot\omega_{n-1}\cdot (1-\varepsilon)$, where $\varepsilon$ is a sufficiently small nonnegative constant and $\omega_n$ is a volume of the unit ball in $R^n$, then in the globular layer $(1+\varepsilon^\delta)^{-1}\le\|x\|\le 1$, $\delta=2^{-n}\cdot(n!)^{-2}$ it is possible to place a parallelepiped symmetric with respect the origin $o$.