Canonical Forms as Dual Volumes

Abstract

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.