On Nonconvexity of the Set of Hypergraphic Sequences

Abstract

In this paper, we prove that 𝑫𝑫 𝒎𝒎 (𝒏𝒏), the set of hypergraphic sequences of all simple hypergraphs ([𝒏𝒏], 𝑬𝑬), where [𝒏𝒏] = {𝟏𝟏, 𝟐𝟐, ⋯ , 𝒏𝒏}, and |𝑬𝑬| = 𝒎𝒎; being a subset of 𝒏𝒏dimensional 𝒎𝒎 + 𝟏𝟏 -valued grid 𝜩𝜩 𝒎𝒎+𝟏𝟏 𝒏𝒏 , is not a convex set in 𝜩𝜩 𝒎𝒎+𝟏𝟏 𝒏𝒏 ; also, we characterize the smallest convex set containing 𝑫𝑫 𝒎𝒎 (𝒏𝒏).