Minoration via mixed volumes and Cover’s problem for general channels

Research output: Contribution to journalArticlepeer-review


We give a complete solution to an open problem of Thomas Cover in 1987 about the capacity of a relay channel in the general discrete memoryless setting without any additional assumptions. The key step in our approach is to lower bound a certain soft-max of a stochastic process by convex geometry methods, which is based on two ideas: First, the soft-max is lower bounded in terms of the supremum of another process, by approximating a convex set with a polytope with bounded number of vertices. Second, using a result of Pajor, the supremum of the process is lower bounded in terms of packing numbers by means of mixed-volume inequalities (Minkowski’s first inequality).

Original languageEnglish (US)
Pages (from-to)315-357
Number of pages43
JournalProbability Theory and Related Fields
Issue number1-2
StatePublished - Jun 2022


  • Convex geometry
  • Empirical process theory
  • High dimensional probability
  • Minkowski’s inequality
  • Multiuser information theory

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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