Sharp estimates for Lebesgue constants

Abstract:

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