On Stability of a Unit Ball in Minkowski Space with Respect to Self-Area

摘要

The main results of the paper are the following two statements. If the length of the unit circle ? B = {|| x || = 1} on Minkowski plane M ² is equal to O ( B ) = 8(1 - e ), 0≤ e ≤ 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 e )^-1 ≤ || x || ≤ 1. If the area of the unit sphere ? B in the Minkowski space Mⁿ , n ≥ 3, is equal to O ( B ) = 2 n ? w _{n -1} ? (1- e ), where e is a sufficiently small nonnegative constant and w _n is a volume of the unit ball in Rⁿ , then in the globular layer (1+ e ^d )^-1 ≤ || x || ≤ 1, d = 2^{- n} ? ( n !)^-2 it is possible to place a parallelepiped symmetric with respect the origin o .