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

Integral Geometry about the visual angle of a convex set

In this paper we deal with a general type of integral formulas of the visual angle, among them those of Crofton, Hurwitz and Masotti, from the point of view of Integral Geometry. The purpose is twofold: to provide an interpretation of these formulas in terms of integrals of densities with respect to the canonical measure in the space of pairs of lines and to give new simpler proofs of them.

关于凸集的周长、面积与视角的若干关系

On some relations between the Perimeter, the Area and the Visual Angle of a Convex Set

该研究探讨平面紧凸集的周长、面积与视角的关系，证明Crofton公式是唯一联系三者的通用公式，给出常宽凸集的等距集无穷远性态刻画，并证实常宽凸集若存在等距圆则必为圆盘。

A note on Hurwitz's inequality

Given a simple closed plane curve $Γ$ of length $L$ enclosing a compact convex set $K$ of area $F$, Hurwitz found an upper bound for the isoperimetric deficit, namely $L^2-4πF\leq π|F_{e}|$, where $F_{e}$ is the algebraic area enclosed by the evolute of $Γ$. In this note we improve this inequality finding strictly positive lower bounds for the deficit $π|F_{e}|-Δ$, where $Δ=L^{2}-4πF$. These bounds involve wether the visual angle of $Γ$ or the pedal curve associated to $K$ with respect to the Steiner point of $K$ or the $\mathcal{L}^{2}$ distance between $K$ and the Steiner disk of $K$. For ea…

On the solid angle of a convex set

Here we analyze three dimensional analogues of the classical Crofton's formula for planar compact convex sets. In this formula a fundamental role is played by the visual angle of the convex set from an exterior point. A generalization of the visual angle to convex sets in euclidian space is the visual solid angle. This solid angle, being an spherically convex set in the unit sphere, has length, area and other geometric quantities to be considered. The main goal of this note is to express invariant quantities of the original convex set depending on volume, surface area and mean curvature integr…