Area function

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The set function on a sphere equal to the area of that part of the convex surface that has spherical image . This definition remains meaningful for general convex surfaces and it gives a totally-additive set function on the ring of Borel sets.


[1] A.D. Aleksandrov, Mat. Sb. , 3 : 1 (1938) pp. 27–44
[2] H. Busemann, "Convex surfaces" , Interscience (1958)


In the article, is the unit sphere in with centre at the origin. If one associates to each point the unit normal vector and shifts this vector so that its base coincides with the origin, then the end point of is a point on . The point is called the spherical image of . The procedure for obtaining the spherical image of a point goes under the name of spherical mapping, cf. Spherical map.

How to Cite This Entry:
Area function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article