Stability of Inequalities in the Dual Brunn-Minkowski Theory
Stability versions are given of several inequalities from E. Lutwak's dual Brunn-Minkowski theory. These include the dual Aleksandrov-Fenchel inequality, the dual Brunn-Minkowski inequality, and the dual isoperimetric inequality. Two methods are used. One involves the application of strong form...
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| Published in | Journal of mathematical analysis and applications Vol. 231; no. 2; pp. 568 - 587 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
San Diego, CA
Elsevier Inc
15.03.1999
Elsevier |
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| Abstract | Stability versions are given of several inequalities from E. Lutwak's dual Brunn-Minkowski theory. These include the dual Aleksandrov-Fenchel inequality, the dual Brunn-Minkowski inequality, and the dual isoperimetric inequality. Two methods are used. One involves the application of strong forms of Clarkson's inequality forLpnorms that hold for nonnegative functions, and the other utilizes a refinement of the arithmetic-geometric mean inequality. A new and more informative proof of the equivalence of the dual Brunn-Minkowski inequality and the dual Minkowski inequality is given. The main results are shown to be the best possible up to constant factors. |
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| AbstractList | Stability versions are given of several inequalities from E. Lutwak's dual Brunn-Minkowski theory. These include the dual Aleksandrov-Fenchel inequality, the dual Brunn-Minkowski inequality, and the dual isoperimetric inequality. Two methods are used. One involves the application of strong forms of Clarkson's inequality forLpnorms that hold for nonnegative functions, and the other utilizes a refinement of the arithmetic-geometric mean inequality. A new and more informative proof of the equivalence of the dual Brunn-Minkowski inequality and the dual Minkowski inequality is given. The main results are shown to be the best possible up to constant factors. |
| Author | Vassallo, S. Gardner, R.J. |
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| Cites_doi | 10.1007/BF02567927 10.2307/2118606 10.2140/pjm.1975.58.531 10.1016/0021-9045(85)90051-6 10.1090/S0002-9947-1994-1201126-7 10.1007/BF00147607 10.2307/2118603 10.1112/S0025579300013565 10.1090/S0002-9939-1990-1015678-5 10.1007/BF00181500 10.1006/aima.1994.1075 |
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| Keywords | Clarkson's inequality Aleksandrov-Fenchel inequality arithmetic-geometric mean inequality isoperimetric inequality geometric tomography Brunn-Minkowski inequality star body convex body dual mixed volume stability Stability Geometric mean Aleksandrov Fenchel inequality Tomography Convex body Brunn Minkowski theory Arithmetic mean Clarkson inequality Brunn Minkonwski inequality Inequality |
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| References | Ann. Math. Schneider (RF19) 1993 Vitale (RF20) 1985; 45 Gaoyong, Zhang, A positive solution to the Busemann-Petty problem in R. J. Gardner, S. Vassallo, Inequalities for dual isoperimetric deficits, Mathematika Gardner (RF4) 1995; 42 Groemer (RF8) 1988; 27 Jakimovski, Russell (RF12) 1981; 3 Mitrinović (RF15) 1970 Lutwak (RF14) 1975; 58 Gardner (RF1) 1994; 342 Schneider (RF18) 1990; 69 Goodey, Groemer (RF7) 1990; 109 Gardner (RF3) 1995 Zhang (RF22) 1994; 140 Mitrinović, Pečarić, Fink (RF16) 1993 Groemer (RF9) 1990; 33 Kober (RF13) 1958; 9 Hewitt, Stromberg (RF11) 1975 Gardner, Volčič (RF6) 1994; 108 Schneider (RF17) 1989; 36 Gardner (RF2) 1994; 140 Groemer (RF10) 1993 Gardner (10.1006/jmaa.1998.6254_RF3) 1995 Mitrinović (10.1006/jmaa.1998.6254_RF15) 1970 Schneider (10.1006/jmaa.1998.6254_RF17) 1989; 36 Schneider (10.1006/jmaa.1998.6254_RF19) 1993 Gardner (10.1006/jmaa.1998.6254_RF4) 1995; 42 10.1006/jmaa.1998.6254_RF5 Groemer (10.1006/jmaa.1998.6254_RF9) 1990; 33 Groemer (10.1006/jmaa.1998.6254_RF10) 1993 Vitale (10.1006/jmaa.1998.6254_RF20) 1985; 45 Groemer (10.1006/jmaa.1998.6254_RF8) 1988; 27 Schneider (10.1006/jmaa.1998.6254_RF18) 1990; 69 10.1006/jmaa.1998.6254_RF21 Gardner (10.1006/jmaa.1998.6254_RF2) 1994; 140 Jakimovski (10.1006/jmaa.1998.6254_RF12) 1981; 3 Mitrinović (10.1006/jmaa.1998.6254_RF16) 1993 Zhang (10.1006/jmaa.1998.6254_RF22) 1994; 140 Goodey (10.1006/jmaa.1998.6254_RF7) 1990; 109 Lutwak (10.1006/jmaa.1998.6254_RF14) 1975; 58 Gardner (10.1006/jmaa.1998.6254_RF1) 1994; 342 Kober (10.1006/jmaa.1998.6254_RF13) 1958; 9 Gardner (10.1006/jmaa.1998.6254_RF6) 1994; 108 Hewitt (10.1006/jmaa.1998.6254_RF11) 1975 |
| References_xml | – year: 1995 ident: RF3 publication-title: Geometric Tomography contributor: fullname: Gardner – volume: 109 start-page: 1103 year: 1990 end-page: 1114 ident: RF7 article-title: Stability results for first order projection bodies publication-title: Proc. Amer. Math. Soc. contributor: fullname: Groemer – volume: 3 start-page: 23 year: 1981 end-page: 27 ident: RF12 article-title: An inequality for the publication-title: C. R. Math. Rep. Acad. Sci. Can. contributor: fullname: Russell – volume: 9 start-page: 452 year: 1958 end-page: 459 ident: RF13 article-title: On the arithmetic and geometric means and on Hölder's inequality publication-title: Proc. Amer. Math. Soc. contributor: fullname: Kober – volume: 42 start-page: 422 year: 1995 end-page: 429 ident: RF4 article-title: Geometric tomography publication-title: Notices Amer. Math. Soc. contributor: fullname: Gardner – volume: 108 start-page: 367 year: 1994 end-page: 399 ident: RF6 article-title: Tomography of convex and star bodies publication-title: Adv. Math. contributor: fullname: Volčič – volume: 140 start-page: 435 year: 1994 end-page: 447 ident: RF2 article-title: A positive answer to the Busemann-Petty problem in three dimensions publication-title: Ann. Math. contributor: fullname: Gardner – volume: 69 start-page: 291 year: 1990 end-page: 300 ident: RF18 article-title: A stability estimate for the Aleksandrov-Fenchel inequality with an application to mean curvature publication-title: Manuscripta Math. contributor: fullname: Schneider – volume: 33 start-page: 117 year: 1990 end-page: 122 ident: RF9 article-title: On an inequality of Minkowski for mixed volumes publication-title: Geom. Dedicata contributor: fullname: Groemer – year: 1975 ident: RF11 publication-title: Real and Abstract Analysis contributor: fullname: Stromberg – volume: 27 start-page: 357 year: 1988 end-page: 371 ident: RF8 article-title: On the Brunn-Minkowski theorem publication-title: Geom. Dedicata contributor: fullname: Groemer – year: 1993 ident: RF19 publication-title: Convex Bodies: The Brunn-Minkowski Theory contributor: fullname: Schneider – year: 1970 ident: RF15 publication-title: Analytic Inequalities contributor: fullname: Mitrinović – start-page: 125 year: 1993 end-page: 150 ident: RF10 article-title: Stability of geometric inequalities publication-title: Handbook of Convexity contributor: fullname: Groemer – volume: 58 start-page: 531 year: 1975 end-page: 538 ident: RF14 article-title: Dual mixed volumes publication-title: Pacific J. Math. contributor: fullname: Lutwak – volume: 45 start-page: 280 year: 1985 end-page: 287 ident: RF20 article-title: metrics for compact, convex sets publication-title: J. Approx. Theory contributor: fullname: Vitale – volume: 36 start-page: 50 year: 1989 end-page: 59 ident: RF17 article-title: Stability in the Aleksandrov-Fenchel-Jessen theorem publication-title: Mathematika contributor: fullname: Schneider – volume: 342 start-page: 435 year: 1994 end-page: 445 ident: RF1 article-title: Intersection bodies and the Busemann-Petty problem publication-title: Trans. Amer. Math. Soc. contributor: fullname: Gardner – volume: 140 start-page: 331 year: 1994 end-page: 346 ident: RF22 article-title: Intersection bodies and the Busemann-Petty inequalities in publication-title: Ann. Math. contributor: fullname: Zhang – year: 1993 ident: RF16 publication-title: Classical and New Inequalities in Analysis contributor: fullname: Fink – volume: 69 start-page: 291 year: 1990 ident: 10.1006/jmaa.1998.6254_RF18 article-title: A stability estimate for the Aleksandrov-Fenchel inequality with an application to mean curvature publication-title: Manuscripta Math. doi: 10.1007/BF02567927 contributor: fullname: Schneider – volume: 42 start-page: 422 year: 1995 ident: 10.1006/jmaa.1998.6254_RF4 article-title: Geometric tomography publication-title: Notices Amer. Math. Soc. contributor: fullname: Gardner – volume: 9 start-page: 452 year: 1958 ident: 10.1006/jmaa.1998.6254_RF13 article-title: On the arithmetic and geometric means and on Hölder's inequality publication-title: Proc. Amer. Math. Soc. contributor: fullname: Kober – volume: 140 start-page: 435 year: 1994 ident: 10.1006/jmaa.1998.6254_RF2 article-title: A positive answer to the Busemann-Petty problem in three dimensions publication-title: Ann. Math. doi: 10.2307/2118606 contributor: fullname: Gardner – volume: 58 start-page: 531 year: 1975 ident: 10.1006/jmaa.1998.6254_RF14 article-title: Dual mixed volumes publication-title: Pacific J. Math. doi: 10.2140/pjm.1975.58.531 contributor: fullname: Lutwak – ident: 10.1006/jmaa.1998.6254_RF5 – volume: 45 start-page: 280 year: 1985 ident: 10.1006/jmaa.1998.6254_RF20 article-title: Lpmetrics for compact, convex sets publication-title: J. Approx. Theory doi: 10.1016/0021-9045(85)90051-6 contributor: fullname: Vitale – volume: 342 start-page: 435 year: 1994 ident: 10.1006/jmaa.1998.6254_RF1 article-title: Intersection bodies and the Busemann-Petty problem publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-1994-1201126-7 contributor: fullname: Gardner – volume: 33 start-page: 117 year: 1990 ident: 10.1006/jmaa.1998.6254_RF9 article-title: On an inequality of Minkowski for mixed volumes publication-title: Geom. Dedicata doi: 10.1007/BF00147607 contributor: fullname: Groemer – volume: 140 start-page: 331 year: 1994 ident: 10.1006/jmaa.1998.6254_RF22 article-title: Intersection bodies and the Busemann-Petty inequalities in R4 publication-title: Ann. Math. doi: 10.2307/2118603 contributor: fullname: Zhang – year: 1995 ident: 10.1006/jmaa.1998.6254_RF3 contributor: fullname: Gardner – ident: 10.1006/jmaa.1998.6254_RF21 – start-page: 125 year: 1993 ident: 10.1006/jmaa.1998.6254_RF10 article-title: Stability of geometric inequalities contributor: fullname: Groemer – volume: 36 start-page: 50 year: 1989 ident: 10.1006/jmaa.1998.6254_RF17 article-title: Stability in the Aleksandrov-Fenchel-Jessen theorem publication-title: Mathematika doi: 10.1112/S0025579300013565 contributor: fullname: Schneider – volume: 109 start-page: 1103 year: 1990 ident: 10.1006/jmaa.1998.6254_RF7 article-title: Stability results for first order projection bodies publication-title: Proc. Amer. Math. Soc. doi: 10.1090/S0002-9939-1990-1015678-5 contributor: fullname: Goodey – year: 1993 ident: 10.1006/jmaa.1998.6254_RF19 contributor: fullname: Schneider – volume: 27 start-page: 357 year: 1988 ident: 10.1006/jmaa.1998.6254_RF8 article-title: On the Brunn-Minkowski theorem publication-title: Geom. Dedicata doi: 10.1007/BF00181500 contributor: fullname: Groemer – volume: 3 start-page: 23 year: 1981 ident: 10.1006/jmaa.1998.6254_RF12 article-title: An inequality for theLp-norm related to uniform convexity publication-title: C. R. Math. Rep. Acad. Sci. Can. contributor: fullname: Jakimovski – year: 1970 ident: 10.1006/jmaa.1998.6254_RF15 contributor: fullname: Mitrinović – year: 1975 ident: 10.1006/jmaa.1998.6254_RF11 contributor: fullname: Hewitt – volume: 108 start-page: 367 year: 1994 ident: 10.1006/jmaa.1998.6254_RF6 article-title: Tomography of convex and star bodies publication-title: Adv. Math. doi: 10.1006/aima.1994.1075 contributor: fullname: Gardner – year: 1993 ident: 10.1006/jmaa.1998.6254_RF16 contributor: fullname: Mitrinović |
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| Snippet | Stability versions are given of several inequalities from E. Lutwak's dual Brunn-Minkowski theory. These include the dual Aleksandrov-Fenchel inequality, the... |
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| SubjectTerms | Aleksandrov-Fenchel inequality arithmetic-geometric mean inequality Brunn-Minkowski inequality Clarkson's inequality convex body dual mixed volume Exact sciences and technology geometric tomography isoperimetric inequality Mathematical analysis Mathematics Real functions Sciences and techniques of general use stability star body |
| Title | Stability of Inequalities in the Dual Brunn-Minkowski Theory |
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