Polyhedral or combinatorial model for the manifolds L_{d,1}

Construct an explicit polyhedral or combinatorial model (e.g., given by linear inequalities) that is diffeomorphic to the manifolds L_{d,1}, analogous to the known polyhedral description of L_{d,0} obtained by replacing the inequality ∏_{i=1}^d x_i = ε with ε ≤ ∑_{i=1}^d (1 − x_i).

Background

In the horn-filling construction, the authors introduce a family of manifolds with corners L_{d,0} ⊂ [0,1]^d and L_{d,1} ⊂ [0,1]^{d+2} that serve as building blocks for smoothing and gluing procedures used to fill inner horns in their semisimplicial framework.

For L_{d,0}, there is a simple polyhedral description: one can replace the multiplicative constraint ∏ x_i ≤ ε with an equivalent (diffeomorphic) model defined by the linear inequality ε ≤ ∑ (1 − x_i). This facilitates concrete manipulations and illustrates geometry useful for later constructions.

In contrast, an analogous explicit (polyhedral/combinatorial) description for L_{d,1} is not currently known to the authors. Providing such a model would clarify the geometric structure of L_{d,1} and could simplify or strengthen subsequent arguments that rely on these blocks.