Convex Set of Doubly Substochastic Matrices

2017/11/18 by Deng, Lei, Lin, Qiulin
#05C50 #05C70 #15A23 #15A24 #15B51 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.06818

Abstract

Denote A as the set of all doubly substochastic m × n matrices and let k be a positive integer. Let A_k be the set of all 1/k-bounded doubly substochastic m × n matrices, i.e., A_k \triangleq \E ∈ A: e_i,j ∈ [0, 1/k], ∀ i=1,2,⋯,m, j = 1,2,⋯, n\. Denote B_k as the set of all matrices in A_k whose entries are either 0 or 1/k. We prove that A_k is the convex hull of all matrices in B_k.

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