Non-symmetric discrete logarithmic Minkowski problem: full characterization

Determine a complete characterization of the cone-volume set C_(U) for arbitrary U ∈ 𝒰(n,m) and establish necessary and sufficient conditions under which a finite discrete Borel measure μ(U,γ) = ∑_{i=1}^m γ_i δ_{u_i} with γ ∈ ℝ^m_{>0} is the cone-volume measure of some polytope P(U,b) = { x ∈ ℝ^n : U^⊤ x ≤ b } with b ∈ ℝ^m_{≥0} and vol(P(U,b)) = 1; that is, solve the non-symmetric discrete logarithmic Minkowski existence problem.

Background

The paper studies the cone-volume set C_(U), consisting of normalized cone-volume vectors γ(U,b) arising from polytopes P(U,b) defined by a matrix U whose columns are unit normals with positive hull ℝ^n. In the symmetric setting U^s = (u_1,…,u_{m'},−u_1,…,−u_{m'}) and γ^s_i = γ^s_{i+m'}, Bőróczky–Lutwak–Yang–Zhang (BLYZ) proved a complete characterization: γ lies in the relative interior of the subspace concentration polytope P_(U^s) intersected with the symmetry constraint, yielding necessary and sufficient conditions.

Beyond the symmetric case, Chen–Li–Zhu established an inclusion relint P_(U) ⊆ C_(U), and Zhu showed that when U is in general position, C_(U) saturates the probability simplex. The present paper proves that C_(U) is always semialgebraic and path-connected, and identifies instances where C_(U) equals P_(U) (precisely for parallelepipeds). However, a full description of C_(U) and a complete set of necessary and sufficient conditions for the non-symmetric discrete logarithmic Minkowski problem remain unknown.