We will show that the quantitative maximal volume entropy rigidity holds on Alexandrov spaces. More precisely, given N, D, there exists epsilon(N, D) > 0, such that for epsilon = -1, diam(X) = N - 1 - epsilon, then X is Gromov-Hausdorff close to a hyperbolic manifold. This result extends the quantitive maximal volume entropy rigidity provided by Chen, Rong, and Xu J. Differential Geom. 113 (2019), pp. 227-272 to Alexandrov spaces. And we will also give a quantitative maximal volume entropy rigidity for RCD*-spaces in the non-collapsing case.
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