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Geometric inequalities and inclusion measures of convex bodies

Published online by Cambridge University Press:  26 February 2010

Gaoyong Zhang
Gaoyong Zhang
Affiliation:
Department of Mathematics, Temple University, Philadelphia, PA 19122, U.S.A.
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In this paper, we will denote by convex figure a compact convex subset of the n-dimensional Euclidean space ℝn, and by convex body a convex figure with non-empty interior. The principal kinematic formula in integral geometry gives the measure of the set of congruent convex bodies intersecting with a fixed convex body. Specifically, let K, L be two convex bodies in ℝn and G(n) the group of special motions in ℝn. Each element, g: ℝn → ℝn, of G(n) can be represented by

where b∈ℝn and e is an orthogonal matrix of determinant 1. Let μ be the Haar measure on G(n) normalized as follows: Let μ:ℝn × SO(n) → G(n) be defined by φ(t, e)x = ex + t, xεℝn, where SO(n) is the rotation group of ℝn. If v is the unique invariant probability measure on SO(n), η is the Lebesgue measure on ℝn, then μ is chosen as the pull back measure of η⊗v under φ−1. If Wi(K), Wi(L) are the quermassintegrals of K, L, i= 0, 1,…, n, the principal kinematic formula states that

where ωn is the volume of the unit n–ball.

Type
Research Article
Information
Mathematika , Volume 41 , Issue 1 , June 1994 , pp. 95 - 116
Copyright
Copyright © University College London 1994

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References

1.Alexandrov, A. D.. On the theory of mixed volumes of convex bodies. Mat. Sb., 3, 45 (1938), 28–46.Google Scholar
2.Ball, K.. Volume ratios and a reverse isoperimetric inequality. J. London Math. Soc, 44 (1991), 351–359.CrossRefGoogle Scholar
3.Bol, G.. Beweis einer Vermutung von H. Minkowski. Abh. Math. Sem. Univ. Hamburg, 15 (1943), 37–56.CrossRefGoogle Scholar
4.Burago, Y. D. and Zalgaller, V. A.. Geometric Inequalities (Springer, Berlin, 1980).Google Scholar
5.Busemann, H.. Volume in terms of concurrent cross-sections. Pacific J. Math., 3 (1953), 1–12.CrossRefGoogle Scholar
6.Busemann, H.. A theorem on convex bodies of the Brunn–Minkowski type. Proc. Nat. Acad. Sci. U.S.A., 35 (1949), 27–31.CrossRefGoogle ScholarPubMed
7.Busemann, H. and Petty, C. M.. Problems on convex bodies. Math. Scand., 4 (1956), 88–94.Google Scholar
8.Chakerian, G. D. and Groemer, H.. Convex bodies of constant width. In Convexity and its Applications, Gruber, P. M. and Wills, J. M., Eds (Birkhauser, Basel, 1983), 49–96.Google Scholar
9.Diskant, V. I.. A generalization of Bonnensen's inequalities. Soviet Math. DokL, 6, 14 (1973), 1728–1731.Google Scholar
10.Gardner, R.. Intersection bodies and the Busemann–Petty problem. To appear.Google Scholar
11.Gordon, Y., Meyer, M. and Reisner, S.. Zonoids with minimal volume product–A new proof, Proc. Amer. Math. Soc, 104 (1988), 273–276.Google Scholar
12.Grinberg, E.. Isoperimetric inequalities and identities for k–dimensional cross–sections of convex bodies. Math. Ann., 291 (1991), 75–86.CrossRefGoogle Scholar
13.Hadwiger, H.. Uberdeckung ebener Bereiche durch Kreise und Quadrate. Comment. Math. Helv., 13 (1941), 195–200.Google Scholar
14.Hadwiger, H.. Gegenseitige Bedeckbarkeit zweier Eibereiche und Isoperimetrie. Vierteljschr. Naturforsch. Gesellsch. Zurich, 86 (1941), 152–156.Google Scholar
15.Leichtweiss, K.. Konvexe Mengen (Springer, Berlin, 1980).Google Scholar
16.Leichtweiss, K.. Geometric convexity and differential geometry. In Convexity and its Applications, Gruber, P. M. and Wills, J. M., Eds. (Birkhauser, Basel, 1983), 163–169.Google Scholar
17.Lutwak, E.. Mixed projection inequalities. Trans. Amer. Math. Soc, 287 (1985), 91–106.Google Scholar
18.Lutwak, E.. On quermassintegrals of mixed projection bodies. Geometriae Dedicata, 33 (1990), 51–58.CrossRefGoogle Scholar
19.Lutwak, E.. On a conjectured projection inequality of Petty. Contemporary Math., 113 (1990), 171–182.CrossRefGoogle Scholar
20.Lutwak, E.. Inequalities for Hadwiger's harmonic quermassintegrals. Math. Ann., 280 (1988), 165–175.CrossRefGoogle Scholar
21.Lutwak, E.. A general isepiphanic inequality. Proc. Am. Math. Soc, 90 (1984), 415–421.CrossRefGoogle Scholar
22.Lutwak, E.. Intersection bodies and dual mixed volumes. Adv. Math., 71 (1988), 232–261.CrossRefGoogle Scholar
23.Lutwak, E.. Intersection bodies and generalized intersection bodies. To appear.Google Scholar
24.Martini, H.. Some characterizing properties of the simplex. Geometriae Dedicata, 29 (1989), 1–6.CrossRefGoogle Scholar
25.Milman, V. D. and Pajor, A.. Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normal n–dimensional space. In Geometric Aspects of Functional analysis. Lecture Notes in Math., 1376 (1989), 64–104.Google Scholar
26.Petty, C. M.. Projection bodies. Proc. Coll. Convexity, (Copenhagen, 1965) (Kobenhavns Univ. Mat. Inst., 1967), 234–241.Google Scholar
27.Petty, C. M.. Isoperimetric problems. Proc. Conf. on Convexity and Combinatorial Geometry, Univ. of Oklahoma, June 1971, (1972), 26–41.Google Scholar
28.Petty, C. M.. Affine isoperimetric problems. Ann. N.Y. Acad. Sci., 440 (1985), 113–127.CrossRefGoogle Scholar
29.Reisner, S.. Random polytopes and the volume–product of symmetric convex bodies. Math. Scand., 57 (1985), 386–392.CrossRefGoogle Scholar
30.Reisner, S.. Zonoids with minimal volume–product. Math. Z., 192 (1986), 339–346.CrossRefGoogle Scholar
31.Ren, D.. Generalized support function and its applications. Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, 1367–1378.Google Scholar
32.Ren, D.. Two topics in integral geometry. Proceedings of the 1981 Symposium on Differential Geometry and Differential Equations (Shanghai-Hefei) (Science Press, Beijing, 1984), 309–333.Google Scholar
33.Ren, D.. An Introduction to Integral Geometry (Science and Technology Press, Shanghai, 1988).Google Scholar
34.Rogers, C. A. and Shephard, G. C.. The difference body of a convex body. Arch. Math., 8 (1957), 220–233.CrossRefGoogle Scholar
35.Santalό, L. A.. Un invariante afln para los cuerpos convexos del espacio de n dimensiones. Portugal. Math., 8 (1949), 155–161.Google Scholar
36.Santalό, L. A.. On the measure of line segments entirely contained in a convex body. Aspects of Mathematics and its Applications, Barroso, I. A. ed. (Elsevier Science Publishers B.V., 1986).Google Scholar
37.Santalό, L. A.. Integral Geometry and Geometric Probability (Addison-Wesley, 1976).Google Scholar
38.Schneider, R.. Geometric inequalities for Poisson processes of convex bodies and cylinders. Results in Math. 11 (1987), 165–185.Google Scholar
39.Schneider, R.. Random hyperplanes meeting a convex body. Z. Wahr. verw. Gebiete, 61 (1982), 379–387.Google Scholar
40.Zhang, G.. A sufficient condition for one convex body containing another. Chin. Ann. of Math., 9B 4 (1988), 447–451.Google Scholar
41.Zhang, G.. Integral geometric inequalities. Ada Math. Sinica, 34 (1991), 72–90.Google Scholar
42.Zhang, G.. Restricted chord projection and affine inequalities. Geometriae Dedicata, 39 (1991), 213–222.Google Scholar
43.Zhang, G.. Characterizations of zonoids and intersection bodies. To appear.Google Scholar
44.Zhang, G.. Intersection bodies and the Busemann–Petty inequalities in R4. To appear.Google Scholar
45.Zhang, G.. Intersection bodies and the four–dimensional Busemann–Petty problem. Duke Math. J., 71 (1993), 233–240.Google Scholar