For a measure $\mu$ supported on a convex body $K$, what are the conditions on $\mu$ and $K$ to satisfy a Log-Sobolev inequality of the form: $$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\int f^{2}\,d\mu\right)\leq \frac{2}{c}\int |\nabla f|^{2}d\mu,$$ for all $f\in C^{1}(K)$ ?

I would appreciate any reference or a solution for the above question.