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 ${\displaystyle n}$ dimensions, one has to consider ${\displaystyle (n-1)}$-dimensional hyperplanes perpendicular to a given direction ${\displaystyle {\hat {n}}}$ in ${\displaystyle S^{n-1}}$, where ${\displaystyle S^{n}}$ is the n-sphere (the surface of a ${\displaystyle (n+1)}$-dimensional sphere). The "width" of a body in a given direction ${\displaystyle {\hat {n}}}$ 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 ${\displaystyle {\hat {n}}}$ in ${\displaystyle S^{n-1}}$.

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