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In 1970, Schneider introduced the higher-order difference body and theassociated Rogers-Shephard inequality. Recently, Haddad, Langharst, Putterman,Roysdon and Ye expanded the concept to a burgeoning higher-orderBrunn-Minkowski theory. In 1991, Zhang introduced mean zonoids of a convexbody, which was extended to the Firey-Brunn-Minkowski theory setting by Xi, Guoand Leng in 2014. In this note, we extend these $L^p$ mean zonoids to thehigher-order setting and establish the associated isoperimetric inequality.

General Higher Order $L^p$ Mean Zonoids