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We study the (non-)attainability of the best constant $C_N(\\Omega)$ in several cases. We provide sufficient conditions that assure $C_N(\\Omega) > C_N(B_R)$ and $C_N(\\Omega)$ is attained, here $B_R$ is the $N$-dimensional ball with center the origin and radius $R$. 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We study the (non-)attainability of the best constant $C_N(\\Omega)$ in several cases. We provide sufficient conditions that assure $C_N(\\Omega) > C_N(B_R)$ and $C_N(\\Omega)$ is attained, here $B_R$ is the $N$-dimensional ball with center the origin and radius $R$. 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