Lesson plan
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 n dimensions, one has to consider (n-1) -dimensional hyperplanes perpendicular to a given direction \hat{n} in S^{n-1} , where S^n is the n-sphere (the surface of a (n+1) -dimensional sphere). The "width" of a body in a given direction \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 \hat{n} in 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 (here, points denote elements of \mathbb{R}^n ). The support function of body B is defined as : h_B(n)=\max\{ \langle n,x\rangle |x \in B \} where n is a direction and \langle,\rangle denotes the usual inner product on \mathbb{R}^n . The mean width is then : b(B)=\frac{1}{S_{n-1}} \int_{S^{n-1}} h_B(\hat{n})+h_B(-\hat{n}), where S_{n-1} is the (n-1) -dimensional volume of S^{n-1} . 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).
Article summary
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 dimensions, one has to consider (n-1) -dimensional hyperplanes perpendicular to a given direction \hat{n} in S^{n-1} , where S^n is the n-sphere (the surface of a (n+1) -dimensional sphere). The "width" of a body in a given direction \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 \hat{n} in 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 (here, points denote elements of \mathbb{R}^n ). The support function of body B is defined as : h_B(n)=\max\{ \langle n,x\rangle |x \in B \} where n is a direction and \langle,\rangle denotes the usual inner product on \mathbb{R}^n . The mean width is then : b(B)=\frac{1}{S_{n-1}} \int_{S^{n-1}} h_B(\hat{n})+h_B(-\hat{n}), where S_{n-1} is the (n-1) -dimensional volume of S^{n-1} . 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).
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