  We establish variant Khintchine inequalities on normed spaces of Hanner type
and cotype, in which the Rademacher distribution corresponding to classical
Khintchine inequality is replaced by general symmetric distributions. The proof
involves the $p$-barycenter and Birkhoff's ergodic theorem. More importantly,
by employing these Khintchine inequalities, we get some lower bounds for
Banach-Mazur distance between $l^p$-ball and a general centrally symmetric
convex body.
