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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Literatur
W. Blaschke, Vorlesungen über Integralgeometrie. VEB Deutscher Verlag der Wissenschaften, Berlin, 1955, 3. Auflage.
P. McMullen, List of problems (Problem 3). Oberwolfach, Mai 1974.
H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957.
J. Raetz, Ueber Inhalt und Oberfläche von Kugel und Zylinder. Math-Phys. Semesterber. 15(1968), 88–93.
G. Ewald, D. G. Larman und C. A. Rogers, The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space. Mathematika 17 (1970), 1–20.
H. Federer, Geometric measure theory. Springer-Verlag, Berlin-Heidelberg-New York, 1969.
C. A. Rogers, Hausdorff measures. Cambridge University Press, London-New York, 1970.
H. Federer, Hausdorff measure and Lebesgue area. Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 90–94.
S. Saks, Theory of the integral. Warschau, 1937.
H. Federer, The area of a nonparametric surface. Proc. Amer. Math. Soc. 11 (1960), 436–439.
I. P. Natanson, Theorie der Funktionen einer reellen Veränderlichen. Akademie-Verlag, Berlin, 1954.
R. N. Tompson, Areas of k-dimensional nonparametric surfaces in k + 1 space. Trans. Amer. Math. Soc. 77 (1954), 374–407.
H. Federer, On Lebesgue area. Ann. of Math. 61 Nr. 2 (1955), 289–353.
L. I. Alpert und L. V. Toralballa, An elementary definition of surface area in En+1 for smooth surfaces. Pacific J. Math. 40 Nr. 2 (1972), 261–268.
J. Serrin, On the area of curved surfaces. Amer. Math. Monthly 68 (1961), 435–440.
S. Eilenberg und N. Steenrod, Foundations of algebraic topology. Princeton University Press, Princeton, 1952.
P. McMullen und G. C. Shephard, Convex polytopes and the upper bound conjecture. Cambridge University Press, London-New York, 1971.
H. Federer, The (ϕ, k) rectifiable subsets of n space. Trans. Amer. Math. Soc. 62 (1947), 114–192.
F. Hausdorff, Dimension und äusseres Mass. Math. Ann. 79 (1918), 157–179.
H. Busemann, The geometry of geodesics. Academic Press, New York, 1955.
H. Hadwiger, Lineare, polyedrisch-limitierbare Eikörperfunktionale. Unveröffentlichtes Manuskript, Bern, 1976.
D. G. Larman und P. Mani, Almost all shadow boundaries are almost smooth. In Vorbereitung.
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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