Your result, including your proposed generalization and the further generalization to surface area in higher dimensions, is an immediate consequence of Cauchy's surface area formula. This states that, up to a constant depending only on the dimension, the surface area of a convex body is the average of the areas of its 1-codimensional projections. I don't know a good reference in a web page, but see for example Klein Klain and Rota's book.
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Your result, including your proposed generalization and the further generalization to surface area in higher dimensions, is an immediate consequence of Cauchy's surface area formula. This states that, up to a constant depending only on the dimension, the surface area of a convex body is the average of the areas of its 1-codimensional projections. I don't know a good reference in a web page, but see for example Klein and Rota's book. |
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