Firey-Shephard problems for homogeneous measures
In recent years, the Brunn-Minkowski theory and its dual have been developed for general measures. A natural question is whether one can expect an affirmative answer to the analogue Firey-Shephard problem for measures. The present paper is concerned with the Firey-Shephard problem for general measures. To this end, some notions of the \(L_p\)-Brunn-Minkowski theory for general measures are introduced, and in particular, the concept of Firey-projection function for a \(\mu\)-measure, which is an extension of the \(L_p\)-projection body. The Firey-Shephard problem for measures is considered and is connected to the \(L^p\)-projection body. Using this connection, a negative answer to the Firey-Shephard problem for homogeneous measures for all \(p>1\), \(p\neq n\) and \(n\geq 2\) is given. A measure comparison theorem is put forward.
scientific article; zbMATH DE number 6802301
scientific article