The classical Bonnet-Myers theorem states that positive section curvature or positive Ricci curvature controls the scale of a manifold. In general, we do not have such kind of theorem for positive scalar curvature. However, Gromov raised an inequality on cubes that positive scalar curvature controls the scale of the cube in some sense. Gromov proved this inequality using the method of minimal surfaces for lower dimensions. In this talk, I will discuss the cube inequality and its index theoretical proof for all dimensions. This is a joint work with Zhizhang Xie and Guoliang Yu.

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