We sketch how Loomis–Whitney follows from this: Indeed, let X be a uniformly distributed random variable with values in A and so that each point in A occurs with equal probability. Then (by the further properties of entropy mentioned above) Η(X) = log|A|, where |A| denotes the cardinality of A. Let Si = {1, 2, …, i−1, i+1, …, d}. The range of ( X j ) j ∈ S i {\displaystyle (X_{j})_{j\in S_{i}}} is contained in Pi(A) and hence H [ ( X j ) j ∈ S i ] ≤ log ⁡ | P i ( A ) | {\displaystyle \mathrm {H} [(X_{j})_{j\in S_{i}}]\leq \log |P_{i}(A)|} .