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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 n {\displaystyle n} dimensions, one has to consider {\displaystyle } -dimensional hyperplanes perpendicular to a given direction n ^ {\displaystyle {\hat {n}}} in S n − 1 {\displaystyle S^{n-1}} , where S n {\displaystyle S^{n}} is the n-sphere {\displaystyle } -dimensional sphere].The "width" of a body in a given direction n ^ {\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 mean width is the average of this "width" over all n ^ {\displaystyle {\hat {n}}} in S n − 1 {\displaystyle S^{n-1}}.
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 n {\displaystyle n} is a direction and ⟨ , ⟩ {\displaystyle \langle ,\rangle } denotes the usual inner product on R n {\displaystyle \mathbb {R} ^{n}}. The mean width is then
where S n − 1 {\displaystyle S_{n-1}} is the {\displaystyle } -dimensional volume of S n − 1 {\displaystyle S^{n-1}}.Note, that the mean width can be defined for any body , but it is mostuseful for convex bodies.