# Area function

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.

#### 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.

**How to Cite This Entry:**

Area function.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Area_function&oldid=32073