  Recently, Gardner, Hug and Weil have introduced the Orlicz-Brunn-Minkowski
theory: a general framework, additions, and inequalities. Following this, in
the paper we consider Orlicz dual Brunn-Minkowski theory. We introduce Orlicz
hormonic Blaschke addition which is an extension of the Lp hormonic Blaschke
addition and L p radial Minkowski addition, respectively. Inequalities of dual
Minkowski and Brunn-Minkowski type are obtained for the Orlicz hormonic
Blaschke addition. The new Orlicz dual Brunn-Minkowski inequality implies the
dual and L p -dual Brunn-Minkowski inequalities, respectively. New Orlicz dual
Minkowski inequality implies the L p -dual Minkowski inequality. One of these
has connections with the conjectured log-Brunn-Minkowski inequality of Lutwak,
Yang, and Zhang, and in fact we show a log dual Minkowski inequality. Finally,
we introduce the concept of Orlicz dual projection body and an inequality
similar to Orlicz projection body is established.
