Stability of Inequalities in the Dual Brunn-Minkowski Theory

• Published in: Journal of mathematical analysis and applications Vol. 231; no. 2; pp. 568 - 587
• Main Authors: Gardner, R.J., Vassallo, S.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 15.03.1999; Elsevier
• Subjects: 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; 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
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1006/jmaa.1998.6254

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.