Shannon's inequality on non-collapsing RCD spaces

We prove the Shannon’s inequality on non-collapsing $\mathsf{R}\mathsf{C}\mathsf{D}(0,N)$ spaces. In the proof, we use the characterization of the ${\mathsf{E}\mathsf{V}\mathsf{I}}_{0,N}$-gradient flow of the relative entropy and the infinitesimal behavior of the heat kernel. Also we have a cone rigidity result. As an application, we have the so-called uncertainty principle inequality on such spaces.