Abstract
It is proved that if ∏*K is the polar projection body of a convex body K in R n, then the volumes of K and ∏*K satisfy the inequality
with equality if and only if K is a simplex. A new zonoid, called the mean zonoid, is defined and some inequalities which characterize the simplices are also proved.
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References
Ball, K., ‘Volume ratios and a reverse isoperimetric inequality’, J. London Math. Soc. (to appear).
Eggleston, H. G., ‘Note on a conjecture of L. A. Santaló’, Mathematika 8 (1961), 63–65.
Lutwak, E., ‘Mixed projection inequalities’, Trans. Amer. Math. Soc. 1 (1985), 91–106.
Martini, H., ‘Some characterizing properties of the simplex’, Geom. Dedicata 29 (1989), 1–6.
Milman, V. D. and Pajor, A., ‘Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space’, Geometric Aspects of Functional Analysis, Lecture Notes in Math., 1376 (1989), 64–104.
Petty, C. M., ‘Centroid surfaces’, Pacific J. Math. 11 (1961), 1535–1547.
Petty, C. M., ‘Projection bodies’, Proc. Colloq. on Convexity, 1967, pp. 234–241.
Delin, Ren and Gaoyong, Zhang, ‘On a type of integral geometric method in the study of geometric probability’, DD6 Symposium, Shanghai, 1985.
Santaló, L. A., Integral Geometry and Geometric Probability, Addison-Wesley, 1976.
Schneider, R., ‘Inequalities for random flats meeting a convex body’, J. Appl. Prob. 22 (1985), 710–716.
Schneider, R., ‘Geometric inequalities for Poisson processes of convex bodies and cylinders’, Results in Math. 11 (1987), 165–185.
Schneider, R., ‘Random hyperplanes meeting a convex body’, Z. Wahrscheinlichkeitstheorie verw. Gebiete 61 (1982), 379–387.
Pfiefer, R. E., ‘The extrema of geometric mean values’, Thesis, Univ. of California, Davis, 1982.
Gaoyong, Zhang, ‘Integral geometric inequalities’, Acta Math. Sinica (Chinese), 34, 1 (1991).
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Zhang, G. Restricted chord projection and affine inequalities. Geom Dedicata 39, 213–222 (1991). https://doi.org/10.1007/BF00182294
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DOI: https://doi.org/10.1007/BF00182294