Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points?

Given a convex body K,, such that t K=-K, is there a point; such that  $diam(\partial K)=d(x,-x)$

I am curious about this question when the metric (on $\partial K$) is the induced (pseudoriemmanian) metric from euclidean space, but mostly in the case of a polytope, with the graph metric on the one skeleton. In the discrete case I do not care too much about constants.