A note on the conditional moments of a multivariate normal distribution confined to a convex set

Let Y be an N([mu], [Sigma]) random variable on Rm, 1 <= m <= [infinity], where [Sigma] is positive definite. Let C be a nonempty convex set in Rm with closure . Let (·,-·) be the Eculidean inner product on Rm, and let [mu]c be the conditional expected value of Y given Y [set membership, variant] C. For v [set membership, variant] Rm and s >= 0, let [beta]s(v) be the expected value of (v, Y) - (v, [mu])s and let [gamma]s(v) be the conditional expected value of (v, Y) - (v, [mu]c)s given Y [set membership, variant] C. For s >= 1, [gamma]s(v) < [beta]s(v) if and only if , and [gamma]s(v) < [beta]s(v) for all v [not equal to] 0 if and only if for any v [set membership, variant] Rm such that v [not equal to] 0.