The Average Projected Area Theorem - Generalization to Higher Dimensions

Details The Average Projected Area Theorem - Generalization to Higher Dimensions The Average Projected Area Theorem - Generalization to Higher Dimensions

2011-09-03

arxiv.org/abs/1109.0595v3

Mathematics

Differential Geometry

9 pages, 1 figure, 1 table, v1 submitted to The American Mathematical Monthly, v2 with revisions to presentation but results u

Scientific paper

It is well known that in 3-d the average projected area of a convex solid is 1/4 the surface area. In this work, we generalize this theorem to higher dimensions by computing the analogous ratio as a function of the dimension. We prove a method for calculating this ratio in higher dimensions. We use this method to obtain both a recursion relation for this ratio from dimension d to d+1 and an explicit formula for it. We discuss the limiting behavior as the dimension becomes infinite and also mention possible application of the theorem as a test of the dimensionality of space.

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