On the integral formulas of Crofton and Hurwitz relative to the visual angle of a convex set

TL;DR

This paper unifies and extends integral formulas related to the visual angle of convex sets, providing new bounds and a versatile approach that generalizes previous results by Crofton, Hurwitz, Masotti, and Santaló.

Contribution

It introduces a unified integral formula framework for visual angles of convex sets, enabling the integration of new functions and generalizing existing bounds.

Findings

Unified approach to integral formulas for visual angles.

Derived new bounds for integrals related to convex sets.

Generalized previous results by Santaló and others.

Abstract

We provide a unified approach that encompasses some integral formulas for functions of the visual angle of a compact convex set due to Crofton, Hurwitz and Masotti. The basic tool is an integral formula that also allows us to integrate new functions of the visual angle. As well we establish some upper and lower bounds for the considered integrals, generalizing in particular those obtained by Santal\'o for Masotti's integral.