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.

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 planes only intersect with the boundary of the body).

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

The support function of body B is defined as where

denotes the usual inner product on

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).

The mean width of a line segment L is the length (1-volume) of L. 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.

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

Similar relations can be given between the other measures and the generalizations of the mean curvature, also for other dimensions .

[1] As the integral over the mean curvature is typically much easier to calculate than the mean width, this is a very useful result.

The mean width is usually mentioned in any good reference on convex geometry, for instance, Selected topics in convex geometry by Maria Moszyńska (Birkhäuser, Boston 2006).

The relation between the mean width and the mean curvature is also derived in that reference.

The application of the mean width as one of the measures featuring in Hadwiger's theorem is discussed in Beifang Chen in "A simplified elementary proof of Hadwiger's volume theorem."