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

Richard J. Gardner

• 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.