I am studying about Tukey's depth and the characterization of distributions by Tukey's depth and I don't know the proofs of following results:

(a) $Q_a=(x \in \mathbb{R}^d:D(x) \geq \alpha)$ are closed convex sets, where $D(.)$ is the Tukey's depth.

(b) If $F\neq G$ are absolutely continuous probability distributions on $\mathbb{R}^d$ then the set of points $((D_F(x), D_G(x)):x\in\mathbb{R}^d)$ in $\mathbb{R}^2$ have non zero Lebesgue measure.

(c)Let $$C_p=\bigcap_t\{R(t):\mathbb{P}(R(t))\geq p\}$$, where $R(t)=\{x\in \mathbb{R}^d:D(x)>t\}$. Then if $F$ id absolutley continuous and the corresponding densoty function is non - zero everywhere then $C_p=R(t_p)$, where $t_p$ is such that $\mathbb{P}(x\in\mathbb{R}^d:D(x) \geq t_p)=p$.

Can someone help me with these proofs?



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