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We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in$S^{2^k-1}$, calculating the Euler class obstruction. Among the consequencesare: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbbR^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$by $(Z_2)^{k+1}$-symmetric convex fans.

Knaster's problem for $(Z_2)^k$-symmetric subsets of the sphere $S^{2^k-1}$