Combinatorial analysis demonstrates an upper bound of 6^(d/2) facets in d-dimensional twinned chain polytopes, highlighting a partial resolution to Nill's conjecture for reflexive polytopes.
Twinned chain polytopes form a broad class of non-centrally symmetric reflexive polytopes and exhibit intriguing structures. In the present paper, we show that the number of facets of d -dimensional twinned chain polytopes is at most 6d/2 6 d / 2 . In case d is even, the equality holds if and only if the polytope is isomorphic to a free sum of d /2 copies of del Pezzo polygons. This result contributes a partial answer to Nill’s conjecture: the number of facets of a d -dimensional reflexive polytope is at most 6d/2 6 d / 2 .
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Mori et al. (2026) studied this question.
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