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We introduce several classes of polytopes contained in $[0,1]^n$ and cut outby inequalities involving sums of consecutive coordinates. We show that thenormalized volumes of these polytopes enumerate circular extensions of certainpartial cyclic orders. Among other things this gives a new point of view on aquestion popularized by Stanley. We also provide a combinatorial interpretationof the Ehrhart $h^*$-polynomials of some of these polytopes in terms ofdescents of total cyclic orders. The Euler numbers, the Eulerian numbers andthe Narayana numbers appear as special cases.

Extensions of partial cyclic orders and consecutive coordinate polytopes