Difference between revisions of "Area function"
From Encyclopedia of Mathematics
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| − | The set function on a sphere | + | {{TEX|done}} |
| + | 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. | ||
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====Comments==== | ====Comments==== | ||
| − | In the article, | + | 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|Spherical map]]. |
Latest revision as of 15:31, 1 May 2014
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://encyclopediaofmath.org/index.php?title=Area_function&oldid=32073
Area function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Area_function&oldid=32073
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article