# Is the $L^p$ norm bounded by the $L^\infty$ norm on a bounded space?

Consider the $$L^p$$ norm defined on some bounded open subset $$\Omega\subset\mathbb{R}^n$$ with $$p>1$$. Does there exist a constant $$c$$ such that $$\Vert f\Vert_{L_p(\Omega)} \le c\Vert f\Vert_{L_\infty(\Omega)}$$ for any function $$f\in L_\infty(\Omega)$$?

I think this is similar to this question, but the inequality is reversed and I am considering the norms over a bounded domain.

$$\int_{\Omega}|f|^p \le \int_{\Omega}||f||_{\infty}^p = m(\Omega) ||f||_{\infty}^p$$ hence, taking $$p$$-th roots, $$||f||_p \le m(\Omega)^{1/p} ||f||_{\infty}$$
Finally, $$c=m(\Omega)^{1/p}$$ is optimal, since the inequality becomes an equality for constant function $$f$$.