  In this work we are interested in the Demyanov--Ryabova conjecture for a
finite family of polytopes. The conjecture asserts that after a finite number
of iterations (successive dualizations), either a 1-cycle or a 2-cycle
eventually comes up. In this work we establish a strong version of this
conjecture under the assumption that the initial family contains "enough
minimal polytopes" whose extreme points are "well placed".
