Editing Circumscribed sphere (section)

==Existence and optimality==
When it exists, a circumscribed sphere need not be the [[Smallest-circle problem|smallest sphere containing the polyhedron]]; for instance, the tetrahedron formed by a vertex of a [[cube]] and its three neighbors has the same circumsphere as the cube itself, but can be contained within a smaller sphere having the three neighboring vertices on its equator. However, the smallest sphere containing a given polyhedron is always the circumsphere of the [[convex hull]] of a subset of the vertices of the polyhedron.<ref name="fgk">{{citation
 | last1 = Fischer | first1 = Kaspar
 | last2 = Gärtner | first2 = Bernd
 | last3 = Kutz | first3 = Martin
 | contribution = Fast smallest-enclosing-ball computation in high dimensions
 | doi = 10.1007/978-3-540-39658-1_57
 | pages = 630–641
 | publisher = Springer
 | series = [[Lecture Notes in Computer Science]]
 | title = Algorithms - ESA 2003: 11th Annual European Symposium, Budapest, Hungary, September 16-19, 2003, Proceedings
 | volume = 2832
 | year = 2003| isbn = 978-3-540-20064-2
 | url = http://www.mpi-inf.mpg.de/~mkutz/pubs/FiGaeKu_SmallEnclBalls.pdf
 }}.</ref>

In ''De solidorum elementis'' (circa 1630), [[René Descartes]] observed that, for a polyhedron with a circumscribed sphere, all faces have circumscribed circles, the circles where the plane of the face meets the circumscribed sphere. Descartes suggested that this necessary condition for the existence of a circumscribed sphere is sufficient, but it is not true: some [[bipyramid]]s, for instance, can have circumscribed circles for their faces (all of which are triangles) but still have no circumscribed sphere for the whole polyhedron. However, whenever a [[simple polyhedron]] has a circumscribed circle for each of its faces, it also has a circumscribed sphere.<ref>{{citation|title=Descartes on Polyhedra: A Study of the "De solidorum elementis"|title-link=Descartes on Polyhedra|first=Pasquale Joseph|last=Federico|authorlink=Pasquale Joseph Federico|series= Sources in the History of Mathematics and Physical Sciences|volume=4|publisher=Springer|year=1982|pages=52–53}}</ref>