Suppose C ⊂ R N C \subset {R^N} is a closed convex bounded body containing 0 in its interior. If ∂ C \partial C is sufficiently smooth with strictly positive Gauss curvature at each point, then, denoting by L r , C {L_{r,C}} the Lebesgue constant relative to C C , there exists a constant A > 0 A > 0 such that L r , C ⩾ A r ( N − 1 ) / 2 {L_{r,C}} \geqslant A{r^{(N - 1)/2}} for r r sufficiently large. This complements the known result that there exists a constant B B such that L r , C ⩽ B r ( N − 1 ) / 2 {L_{r,C}} \leqslant B{r^{(N - 1)/2}} for r r sufficiently large.