Two-partition-symmetrical entropy function regions

Consider the entropy function region for discrete random variables X<sub>i</sub>, i ϵ N and partition N into N<sub>1</sub> and N<sub>2</sub> with 0 ≤ |N<sub>1</sub>| ≤ |N<sub>2</sub>|. An entropy function h is called (N<sub>1</sub>, N<sub>2</sub>)-symmetrical if for all A, B ⊂ N, h(A) = h(B) whenever |A ∩ N<sub>1</sub>| = |B ∩N<sub>1</sub>|, i = 1,2. We prove that for |N<sub>1</sub>| = 0 or 1, the closure of the (N<sub>1</sub>, N<sub>2</sub>)-symmetrical entropy function region is completely characterized by Shannon-type information inequalities. Applications of this work include threshold secret sharing and distributed data storage, where symmetry exists in the structure of the problem.

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