arXiv:2306.17067v2 Announce Type: replace-cross Abstract: A sharp upper bound is obtained for the distance covariance of bounded random vectors in terms of the diameters of their supports alone. The bound is attained for all pairs of support diameter constraints and hence provides the maximum distance covariance subject to constraints of this kind. A normalized measure of dependence is derived by dividing distance covariance by the sharp bound when it is positive. Several properties of the resulting quotient are proved, including domination by distance correlation, invariance under similarities, and subadditivity under sums of independent pairs. The quotient is shown to be nonincreasing under the addition of independent noise, a property not shared by distance correlation. The usual sample distance covariance inherits the sharp bound from the population quantity and the associated normalized statistic is strongly consistent.