Editing Mean width (section)

In geometry, the '''mean width''' is a [[Measure (mathematics)|measure]] of the "size" of a body; see [[Hadwiger's theorem]] for more about the available measures of bodies. In <math>n</math> dimensions, one has to consider <math>(n-1)</math>-dimensional hyperplanes perpendicular to a given direction <math>\hat{n}</math> in <math>S^{n-1}</math>, where <math>S^n</math> is the [[n-sphere]] (the surface of a <math>(n+1)</math>-dimensional sphere).
The "width" of a body in a given direction <math>\hat{n}</math> is the distance between the closest pair of such planes, such that the body is entirely in between the two hyper planes (the planes only intersect 
with the boundary of the body).  The mean width is the average of this "width" over all <math>\hat{n}</math> in <math>S^{n-1}</math>.

[[File:Width in dir n for mean width.png|thumb|alt=width in dir n|The definition of the "width" of body B in direction <math>\hat{n}</math> in 2 dimensions.]]

More formally, define a compact body B as being equivalent to set of points in its interior plus the points on the boundary (here, points denote elements of <math>\mathbb{R}^n</math>). The support function of body B is defined as

: <math>h_B(n)=\max\{ \langle n,x\rangle |x \in B \}</math>

where <math>n</math> is a direction and <math>\langle,\rangle</math> denotes the usual inner product on <math>\mathbb{R}^n</math>.  The mean width is then

: <math>b(B)=\frac{1}{S_{n-1}} \int_{S^{n-1}} h_B(\hat{n})+h_B(-\hat{n}),</math>

where <math>S_{n-1}</math> is the <math>(n-1)</math>-dimensional volume of <math>S^{n-1}</math>.
Note, that the mean width can be defined for any body (that is compact), but it is most
useful for convex bodies (that is bodies, whose corresponding set is a [[convex set]]).