Local and equatorial characterization of unit balls of subspaces of \(L_{p}\), \(p>0\) and properties of the generalized cosine transform (Q549788)

An origin symmetric convex body \(K\subset {\mathbb R}^n\) embeds in \(L_p\), \(p>0\) (using a theorem of Levy) if and only if the norm \(\|\cdot\|_K\) defined by \(K\) satisfies \[ \|x\|_K^p =\int_{S^{n-1}} |x\cdot \theta|^p d\mu(\theta),\quad x\in{\mathbb R}^n, \] with a finite positive Borel measure \(\mu\) on the unit sphere \(S^{n-1}\). A question by W. Weil, whether convex bodies that embed in \(L_1\) admit a local equatorial characterization, was answered in the positive for even dimensions \(n\) by \textit{G. Yu. Panina} [Sov. J. Contemp. Math. Anal., Arm. Acad. Sci. 23, No.~4, 91--103 (1988); translation from Izv. Akad. Nauk Arm. SSR, Mat. 23, No.~4, 385--395 (1988; Zbl 0679.52006)] and in the negative for odd dimensions by \textit{F. Nazarov, D. Ryabogin} and \textit{A. Zvavitch} [Adv. Math. 217, No.~3, 1368--1380 (2008; Zbl 1151.52002)]. In the present paper, it is proved that there is no local equatorial characterization of bodies that embed in \(L_p\) in odd dimensions for all \(p\) not even, \(0<p<\infty\). Further, bodies that embed in \(L_p\) for \(p\) odd admit a local equatorial characterization provided that the dimension is even, but are not locally characterizable in general. The proof uses A. Koldobsky's characterization of embeddability in \(L_p\) in terms of Fourier transforms of distributions, together with the Fourier analytic inversion formula for the \(p\)-cosine transform.