Mahalanobis distance ($d_M(x,y)=\sqrt{(x-y{)}^T{\Sigma }^{-1}(x-y)}$)

For a positive-definite covariance matrix, this is Euclidean distance after whitening. It accounts for both scales and correlations and is invariant under consistent nonsingular linear changes of coordinates. Estimating the covariance can be unstable with small samples, and a singular covariance requires restricting to a nonsingular subspace rather than claiming an ordinary metric on the full space.