Integral Identities for the boundary of a convex body

TL;DR

This paper extends classical integral identities and formulas to multidimensional convex bodies, including boundary and interior point cases, enhancing tools for geometric analysis.

Contribution

It generalizes Pleijel, Ambartzumian--Pleijel, Blaschke--Petkantschin, and Z"ahle formulas to higher dimensions and mixed interior-boundary point configurations.

Findings

Multidimensional Pleijel and Ambartzumian--Pleijel identities derived.

Generalized Blaschke--Petkantschin and Z"ahle formulas for convex bodies.

Z"ahle formula adapted for polytopes.

Abstract

We present the multidimensional versions of the Pleijel and Ambartzumian--Pleijel identities. We also obtain the generalization of both the Blaschke--Petkantschin and Z\"ahle formulae considering the case when some points are chosen inside the convex body and some on the boundary. Moreover, a version of the Z\"ahle formula for the polytopes is derived.