The Gini index and the Gini mean difference of a univariate distribution are extended to measure the disparity of a general $d$- variate distribution. We propose and investigate two approaches, one based on the distance of the distribution from itself, the other on the volume of a convex set in $(d+1)$-space, named the lift zonoid of the distribution. When $d=1$, this volume equals the area between the usual Lorenz curve and the line of zero disparity, up to a scale factor. We get two definitions of the multivariate Gini index, which are different (when $d > 1$) but connected through the notion of the lift zonoid. Both notions inherit properties of the univariate Gini index, in particular, they are vector scale invariant, continuous, bounded by $0$ and $1$, and the bounds are sharp. They vanish if and only if the distribution is concentrated at one point. The indices have a ceteris paribus property and are consistent with multivariate extensions of the Lorenz order. Illustrations with data conclude the paper.
