Sharp lower bounds are found for the concentration of a probability distribution as a function of the expectation of any given convex symmetric function φ. In the case φ(x)=(x-c)2, where c is the expected value of the distribution, these bounds yield the classical concentration-variance inequality of Lévy. An analogous sharp inequality is obtained in a similar linear search setting, where a sharp lower bound for the concentration is found as a function of the maximum probability swept out from a fixed starting point by a path of given length.



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URL: https://digitalcommons.calpoly.edu/rgp_rsr/42