2025/09/02 by Elia Mazzucchelli, Mazzucchelli, Elia, Prashanth Raman +1 · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #High Energy Physics - Theory (hep-th) #Holomorphic and Operator Theory #Point processes and geometric inequalities #Probability (math.PR) #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2509.02239
openalex publication_date 2025/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study dual volume representations of canonical forms for positive geometries in projective spaces, expressing their rational canonical functions as Laplace transforms of measures supported on the convex dual of the semialgebraic set. When the measure is non-negative, we term the geometry completely monotone, reflecting the property of its canonical function. We identify a class of positive geometries whose canonical functions admit such dual volume representations, characterized by the algebraic boundary cut out by a hyperbolic polynomial, for which the geometry is a hyperbolicity region. In particular, simplex-like minimal spectrahedra are completely monotone, with representing measures related to the Wishart distribution, capturing volumes of spectrahedra or their boundaries. We explicitly compute these measures for positive geometries in the projective plane bounded by lines and conics or by a nodal cubic, revealing periods evaluating to transcendental functions. This dual volume perspective reinterprets positive geometries by replacing logarithmic differential forms with probability measures on the dual, forging new connections to partial differential equations, hyperbolicity, convexity, positivity, algebraic statistics, and convex optimization.