Conjectured inequality for polytopes with d+2 vertices inscribed in the unit sphere

Establish the inequality for every convex polytope Q ⊂ S^{d−1} with d+2 vertices that ||vol_d(Q)||_{ℓ2}^2 + ε_d(Q) ≤ [(||vol_1(Q)||_{ℓ2}^2 + ||vol_1(Q^o)||_{ℓ2}^2)^d] / [c_d · (d!)^2 · d^d], where c_d = 2(d+2)^{d−1}, Q^o denotes the added edges that complete the edge graph of Q, and ε_d(Q) is a quadratic form satisfying 0 ≤ ε_d(Q) ≤ C · (vol_d(Q))^2 for some constant C. The ℓ2 norm ||vol_d(Q)||_{ℓ2} is taken over the vector of d-dimensional volumes of the simplices in a partition of Q induced by a point p ∈ Int(Q).

Background

Section 5 investigates polytopes in ℝ^d with d+2 vertices inscribed in the unit sphere S^{d−1}. The authors decompose such a polytope Q into a union of simplices via a point p in the interior, and augment the edge graph by adding certain edges denoted Q^o to facilitate frame-based analysis. They derive specific inequalities for low-dimensional cases (quadrilaterals in ℝ^2, pyramids and bipyramids in ℝ^3) using the trace–determinant inequality and the edge frame, expressed through quadratic forms that capture interactions among the simplex volumes in the partition.

Based on these low-dimensional results and a general framework involving a symmetric positive semidefinite matrix Z with uniform row sums, they propose a general inequality for all dimensions. The conjecture specifies an explicit bound involving the ℓ2 norms of edge lengths of Q and the added edges Q^o, as well as a quadratic form ε_d(Q) constrained between 0 and a constant multiple of the squared volume of Q, with a combinatorial constant c_d = 2(d+2)^{d−1} tied to spanning trees of the edge graph.