Difference between revisions of "Brunn-Minkowski theorem"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (moved Brunn–Minkowski theorem to Brunn-Minkowski theorem: ascii title) |
(No difference)
|
Revision as of 18:51, 24 March 2012
Let and
be convex sets in an
-dimensional Euclidean space; let
,
, be the set of points which divide segments with end points at any points of the sets
and
in the ratio
(a linear combination of
and
); and let
be the
-th power root of the volume of the set
. Then
is a concave function of
, i.e. the inequality
![]() |
is valid for all . The function
is linear (and the inequality then becomes an equality) if and only if
and
are homothetic. The Brunn–Minkowski theorem can be generalized to linear combinations of several convex sets. It is used to solve extremal and uniqueness problems. It was discovered by H. Brunn in 1887, and completed and rendered more precise in 1897 by H. Minkowski.
References
[1] | H. Busemann, "Convex surfaces" , Interscience (1958) |
[2] | H. Hadwiger, "Vorlesungen über Inhalt, Oberfläche und Isoperimetrie" , Springer (1957) |
Comments
References
[a1] | K. Leichtweiss, "Konvexe Mengen" , Springer (1979) |
Brunn-Minkowski theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brunn-Minkowski_theorem&oldid=22205