Abstract
We introduce a new parameter, symmetric average, which measures the asymmetry of a given non-degenerated convex body K in 
. Namely, sav(K) = inf
a∈int K
∫
Ka
‖– x‖
Ka
dx/|K|, where |K| denotes the volume of K and Ka
= K – a. We show that for polytopes sav(K) ≤ C ln N, where N is the number of facets of K. Moreover, in general 
and equality in the lower bound holds if and only if K is centrally symmetric. We apply these estimates to provide bounds for covering K by homothets of K ∩ –K.
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