Majorization precursors to supermodularity and subadditivity on the majorization lattice
2026-05-28 • Information Theory
Information Theory
AI summaryⓘ
The authors identify two new relationships, called precursors, that explain why certain mathematical functions behave nicely when combined. These functions, which they call sum-concave functions, show special properties called supermodularity and subadditivity within a structure called the majorization lattice. Using these ideas, the authors prove that popular entropy measures—Tsallis, Rényi, and Shannon entropies—also have these properties. Furthermore, they show these entropies are strictly subadditive and, in the case of Tsallis and Shannon, strictly supermodular on this lattice.
majorization latticesupermodularitysubadditivitysum-concave functionsTsallis entropyRényi entropyShannon entropyconcave functionsentropy measures
Authors
Alexander Stévins, Michael G. Jabbour, Serge Deside, Nicolas J. Cerf
Abstract
We establish two structural majorization relations, which we call precursors, underlying the properties of supermodularity and subadditivity on the lattice induced by majorization. These are precursors in that they immediately imply that all sums of concave functions, which we dub sum-concave functions, are supermodular and subadditive on the majorization lattice. Using these majorization relations, we then show the supermodularity and subadditivity (in the lattice-theoretic sense) of Tsallis entropies (for all $α$) and Rényi entropies (for all $α> 1$), also recovering these properties for the Shannon entropy in the process. We further strengthen these inequalities, showing that: (i) all these entropic functionals are strictly subadditive on the majorization lattice; (ii) Tsallis entropies (and therefore the Shannon entropy as well) are strictly supermodular on the majorization lattice.