Kinematic formulas in convex geometry

Francisco Eduardo Cardoso Leitão Durão Nascimento

Key information

Authors:

Francisco Eduardo Cardoso Leitão Durão Nascimento (Francisco Eduardo Cardoso Leitão Durão Nascimento)

Supervisors:

Sílvia Nogueira da Rocha Ravasco dos Anjos (Sílvia Nogueira da Rocha Ravasco dos Anjos)

Published in

December 9, 2024

Abstract

We present a systematic study of kinematic formulas in convex geometry. We start with the real case, building from the ground up the necessary prerequisites: the invariant measures used in integration, Steiner's Formula and valuations. Known proofs of several kinematic formulas are presented. Their common link is Hadwiger's Characterization Theorem, which gives a description of $SO(n)$-invariant valuations as a finite-dimensional vector space. We also obtain a new result, extending the domain of integration of the principal kinematic formulas from $SO(n)$ to $GL(n,\mathbb{R})$ through the use of a Gaussian probability measure to integrate along the non-compact positive definite matrices obtained through matrix polar decomposition. We then look at kinematic formulas in the complex case. Although there is an identification of $\mathbb{C}^n$ with $\mathbb{R}^{2n}$, it is different from the real case due to the focus on unitary matrices $U(n)$. In the end, using the same method as in $\mathbb{R}^n$, we extend the domain of integration of the principal kinematic formula in $\mathbb{C}^n$ from $U(n)$ to $Sp(2n,\mathbb{R})$, obtaining another new result.