We prove the Shannon's inequality on non-collapsing RCD(0,N) spaces. In the proof, we use the characterization of the EVI0,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.
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Yu Kitabeppu (2024) studied this question.
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