The dual Brunn¿Minkowski theory for bounded Borel sets: dual affine quermassintegrals and inequalities

• Autores: Richard J. Gardner
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 216, Nº 1, 2007, págs. 358-386
• Idioma: inglés
• DOI: 10.1016/j.aim.2007.05.018
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• Resumen

  • Abstract This paper develops a significant extension of E. Lutwak's dual Brunn¿Minkowski theory, originally applicable only to star-shaped sets, to the class of bounded Borel sets. The focus is on expressions and inequalities involving chord-power integrals, random simplex integrals, and dual affine quermassintegrals. New inequalities obtained include those of isoperimetric and Brunn¿Minkowski type. A new generalization of the well-known Busemann intersection inequality is also proved. Particular attention is given to precise equality conditions, which require results stating that a bounded Borel set, almost all of whose sections of a fixed dimension are essentially convex, is itself essentially convex.