Local and equatorial characterization of unit balls of subspaces of , and properties of the generalized cosine transform

In this paper we show that there is no local equatorial characterization of bodies that embed in L p in odd dimensions for all p not even, 0 < p < ∞ . However, bodies that embed in L p for p odd are local equatorially characterizable provided that the dimension is even but not locally characterizable in general. This extends results given in Panina (1988) [13], Goodey and Weil (1992) [5], and Nazarov, Ryabogin and Zvavitch (2008) [12] concerning the local equatorial characterization of zonoids and intersection bodies. These results were for bodies that embed in L 1 and L − 1 respectively.