  We study the periods mapping from the moduli space of real hyperelliptic
curves with marked point on an oriented oval to the euclidean space. The
mapping arises in the analysis of Chebyshev construction used in the
constrained optimization of the uniform norm of polynomials and rational
functions. The decomposition of the moduli space into polyhedra labeled by
planar graphs allows to investigate the global topology of low dimensional
fibers of the periods mapping.
