Sections and Projections of $$L_p$$-Zonoids and Their Polars

Let \(1\le k\le n\). Sharp upper and lower bounds for the volume of k-dimensional projections (or sections) of \(L_p\)-zonoids (or their polars) with even isotropic generating measures are established. As special cases, sharp volume inequalities for k-dimensional sections and projections of \(\ell _p^n\)-balls are recovered. The necessary conditions of equalities are new.