Mean width


In geometry, the mean width is a measure of the "size" of a body; see Hadwiger's theorem for more about the available measures of bodies. In dimensions, one has to consider -dimensional hyperplanes perpendicular to a given direction in, where is the n-sphere.
The "width" of a body in a given direction is the distance between the closest pair of such planes, such that the body is entirely in between the two hyper planes. The mean width is the average of this "width" over all in.
More formally, define a compact body B as being equivalent to set of points in its interior plus the points on the boundary. The support function of body B is defined as
where is a direction and denotes the usual inner product on. The mean width is then
where is the -dimensional volume of.
Note, that the mean width can be defined for any body, but it is most
useful for convex bodies.

Mean widths of convex bodies in low dimensions

One dimension

The mean width of a line segment L is the length of L.

Two dimensions

The mean width w of any compact shape S in two dimensions is p/π, where p is the perimeter of the convex hull of S. So w is the diameter of a circle with the same perimeter as the convex hull.

Three dimensions

For convex bodies K in three dimensions, the mean width of K is related to the average of the mean curvature, H, over the whole surface of K. In fact,
where is the boundary of the convex body and
a surface integral element, is the mean curvature at the corresponding position
on. Similar relations can be given between the other measures
and the generalizations of the mean curvature, also for other dimensions
As the integral over the mean curvature is typically much easier to calculate
than the mean width, this is a very useful result.