Gauss's inequality is for unimodal distributions, concerning distance from the mode.

A similar result is Vysochanskiï–Petunin inequality, which is for the distance from the mean rather than the mode. Chebyshev's inequality generalizes Vysochanskiï–Petunin inequality by concerning distance from the mean without requiring unimodality.

I wonder if there is generalization of Gauss's inequality for distributions not necessarily unimodal?

Thanks!