Postprint version. Published in Probability Theory and Related Fields, Volume 86, Issue 1, March 1, 1990, pages 53-62. Copyright © 1990 Springer. The original publication is available at http://dx.doi.org/10.1007/BF01207513.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
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.