Sharp Estimates for some Integral-Geometric Quantities Related To Transversality, Curvature And Visibility
Silouanos Brazitikos, Dimitris-Marios Liakopoulos
TL;DR
This work refines integral-geometric bounds for the transverse interaction of vector fields on generalized $d$-hypersurfaces by introducing an angular deficit factor $\rho(x)$, yielding angle-sensitive refinements of Finner-type inequalities for $Q_j^p$. It generalises mixed-volume Bézout-type inequalities to zonoids, derives sharp diagonal-case maximizers depending on $p$, and provides sharp bounds for visibility via projection bodies, Loomis–Whitney, Santaló-type and Brascamp–Lieb inequalities in Lewis position. The results establish precise extremisers and constants across all $p\ge1$ and connect transversality, visibility, and affine-invariant geometry in a unified framework. These contributions have implications for multilinear Kakeya problems and affine-invariant measures on surfaces, with a robust toolkit of local-to-global bounds and isotropic-position arguments. Overall, the paper advances quantitative control of geometric interaction quantities through new angular refinements and comprehensive variational analyses.
Abstract
We investigate integral-geometric quantities arising from harmonic analysis which measure visibility and transversality. Motivated by their applications in multilinear Kakeya problems and affine-invariant measures on surfaces, we derive exact lower and upper bounds employing geometric and functional inequalities of convex geometry.