The set function on a sphere $\Omega$ equal to the area $S(E)$ of that part of the convex surface $F$ that has spherical image $E\subset\Omega$. This definition remains meaningful for general convex surfaces and it gives a totally-additive set function on the ring of Borel sets.
|||A.D. Aleksandrov, Mat. Sb. , 3 : 1 (1938) pp. 27–44|
|||H. Busemann, "Convex surfaces" , Interscience (1958)|
In the article, $\Omega$ is the unit sphere in $\mathbf R^3$ with centre at the origin. If one associates to each point $x\in F$ the unit normal vector $n_x$ and shifts this vector $n_x$ so that its base coincides with the origin, then the end point of $n_x$ is a point $x^*$ on $\Omega$. The point $x^*$ is called the spherical image of $x$. The procedure for obtaining the spherical image of a point goes under the name of spherical mapping, cf. Spherical map.
Area function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Area_function&oldid=32073