Convex Hulls of Great Subspheres and Gaussian Minima

This preliminary AI-assisted research manuscript studies convex hulls of finitely many great subspheres of the Euclidean unit sphere and their relation to Gaussian minima. For unit normals ν_1,...,ν_n in Sᵈ⁻¹, it considers the convex hull of the central great subspheres Sᵈ⁻¹ ∩ ν_i^⊥ and establishes a volume–Gaussian identity with a nonnegative correction term. The manuscript also establishes an exact identity for the second intrinsic volume, an active-stratum representation of the correction under signed affine general position, reduction to the span of the normals, and Rn,d=0 when the normals span a subspace of dimension at most two. In dimension three, it gives an explicit formula valid without general position, including multiple ties, in terms of active polygons. As a conditional consequence, the p=2 part of Kunisky's Gaussian-minimum conjecture implies a sharp upper bound for the convex-hull volume, attained by equally spaced planar normal lines. Generative AI, principally OpenAI's ChatGPT, was used to carry out a substantial majority of the mathematical exploration, conjecture formation, proof development, proof revision, and reproducible diagnostic checking underlying the manuscript. cyclemath formulated and directed the investigation, selected and challenged intermediate claims, requested repeated audits and corrections, and prepared the manuscript for public release. The designation "Editor" / "Edited and released by cyclemath" is intended to record that role without representing the mathematical content as solely or primarily the unaided work of cyclemath. The mathematical arguments and novelty assessment have not received independent verification by a subject-matter expert, and the manuscript has not undergone peer review. The manuscript does not prove Kunisky's Gaussian-minimum conjecture, Kunisky's volumetric zone conjecture, or uniqueness of volume maximizers. The cylindrical/sine-polar framework is prior work and is not claimed as new.