Area function
The set function on a sphere 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.
References
[1] | A.D. Aleksandrov, Mat. Sb. , 3 : 1 (1938) pp. 27–44 |
[2] | H. Busemann, "Convex surfaces" , Interscience (1958) |
Comments
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.
Surface function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Surface_function&oldid=44796