Abstract
In this paper the author first introduce a new concept of L p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L p intersection body to L p -mixed intersection body. Inequalities for L p -dual mixed volumes of L p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities.
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Lutwak E. Intersection bodies and dual mixed volumes. Adv Math, 71: 232–261 (1988)
Goodey P, Lutwak E, Weil W. Functional analytic characterizations of classes of convex bodies. Math Z, 222: 363–381 (1996)
Zhang G. A positive solution to the Busemann-Petty problem in ℝ4. Ann Math, 149: 535–543 (1999)
Gardner R. J. A positive answer to the Busemann-Petty problem in three dimensions. Ann Math, 140: 435–447 (1994)
Gardner R J, Koldobsky A, Schlumprecht T. An analytic solution to the Busemann-Petty problem on sections of convex bodies. Ann Math, 149: 691–703 (1999)
Busemann H. Volume in terms of concurrent cross-sections. Pacific J Math, 3: 1–12 (1953)
Campi S. Stability estimates for star bodies in terms of their intersection bodies. Mathematika, 45: 287–303 (1998)
Campi S. Convex intersection bodies in three and four dimensions. Mathematika, 46: 15–27 (1999)
Fallert H, Goodey P, Weil W. Spherical projections and centrally symmetric sets. Adv Math, 129: 301–322 (1997)
Gardner R J. A positive answer to the Busemann-Petty problem in three dimensions. Ann Math, 140: 435–447 (1994)
Gardner R J. On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies. Bull Amer Math Soc, 30: 222–226 (1994)
Gardner R J. Intersection bodies and the Busemann-Petty problem. Trans Amer Math Soc, 342: 435–445 (1994)
Gardner R J. Geometric Tomography. Cambridge: Cambridge University Press, 1995
Gardner R J, Koldobsky A, Schlumprecht T. An analytic solution to the Busemann-Petty problem. C R Acad Sci Paris S I Math, 328: 29–34 (1999)
Goodey P, Weil W. Intersection bodies and ellipsoids. Mathematika, 42: 295–304 (1995)
Grinberg E, Zhang G. Convolutions, transforms, and convex bodies. Proc London Math Soc, 78: 77–115 (1999)
Koldobsky A. A functional analytic approach to intersection bodies. Geom Funct Anal, 10: 1507–1526 (2000)
Koldobsky A. Intersection bodies, positive definite distributions, and the Busemann-Petty problem. Amer J Math, 120: 827–840 (1998)
Koldobsky A. Second derivative test for intersection bodies. Adv Math, 136: 15–25 (1998)
Koldobsky A. Intersection bodies in R4. Adv Math, 136: 1–14 (1998)
Koldobsky A. Intersection bodies and the Busemann-Petty problem. C R Acad Sci Paris S I Math, 325: 1181–1186 (1997)
Ludwig M. Intersection bodies and valuations. Amer J Math, 128: 1409–1428 (2006)
Moszyńska M. Quotient star bodies, intersection bodies, and star duality. J Math Anal Appl, 232: 45–60 (1999)
Zhang G. Intersection bodies and polytopes. Mathematika, 46: 29–34 (1999)
Zhang G. Sections of convex bodies. Amer J Math, 118: 319–340 (1996)
Zhang G. Intersection bodies and the Busemann-Petty inequalities in ℝ4. Ann Math, 140: 331–346 (1994)
Zhang G. Centered bodies and dual mixed volumes. Trans Amer Math Soc, 345: 777–801 (1994)
Zhang G. Intersection bodies and the four-dimensional Busemann-Petty problem. Internat Math Res Notices, 7: 233–240 (1993)
Lutwak E. Mixed projection inequalities. Trans Amer Math Soc, 287: 91–105 (1985)
Lutwak E. Inequalities for mixed projection bodies. Trans Amer Math Soc, 339: 901–916 (1993)
Lutwak E. Volume of mixed bodies. Trans Amer Math Soc, 294: 487–500 (1986)
Lutwak E. Dual mixed volumes. Pacific J Math, 58: 531–538 (1975)
Haberl C, Ludwig M. A characterization of Lp intersection bodies. Internat Math Res Notices, (2006)
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This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)
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Zhao, C. L p -mixed intersection bodies. Sci. China Ser. A-Math. 51, 2172–2188 (2008). https://doi.org/10.1007/s11425-008-0074-3
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DOI: https://doi.org/10.1007/s11425-008-0074-3