Let \(0 \lt \alpha \le 1/2\). We say that a finite point set \(P\) in \(\mathbb R^d\) is \(\alpha\)-split by a hyperplane \(h\) if each of the closed half-spaces determined by \(h\), contains at least \(\alpha|P|\) of the points of \(P\). We further say \(P\) is \(\alpha\)-split by a \(k\)-dimensional flat \(\tau\) if \(P\) is \(\alpha\)-split by any hyperplane through \(\tau\). In the standard notation (which coincides with Tukey depth for \(k = 0\)), the \(k\)-flat \(\tau\) has depth \(\alpha\) with respect to \(P\).