Summary
Minkowski’s quermassintegral W2(K) and the average measure α(K) of the shadow boundaries of a convex body K in Euclidean space En are closely related. In this relationship n -balls B and n-polytopes P respectively appear in a certain sense as extreme bodies. Verifying a conjecture by P. McMullen, we show for smooth K, that 1 = α(B)/β(B)≤α(K)/β(K)≤nωn/πωn−1=α(P)/β(P), where β(K) = (n−1)ωn−1W2(K)/ωn and ωk denotes the volume of the k -dimensional unit ball. Geometrically β(K) represents the average measure of the relative boundaries of the orthogonal projections of K onto hyperplanes. The polyhedral lower semicontinuity of the functional α, which follows essentially from a fundamental additivity property of the Lebesgue area, is a key-result within the proof.
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Steenaerts, P. Mittlere Schattengrenzenlänge konvexer Körper. Results. Math. 8, 54–77 (1985). https://doi.org/10.1007/BF03322658
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DOI: https://doi.org/10.1007/BF03322658