English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We prove a general duality result showing that a Brascamp--Lieb typeinequality is equivalent to an inequality expressing subadditivity of theentropy, with a complete correspondence of best constants and cases ofequality. This open a new approach to the proof of Brascamp--Lieb typeinequalities, via subadditivity of the entropy. We illustrate the utility ofthis approach by proving a general inequality expressing the subadditivityproperty of the entropy on $\R^n$, and fully determining the cases of equality.As a consequence of the duality mentioned above, we obtain a simple new proofof the classical Brascamp--Lieb inequality, and also a fully explicitdetermination of all of the cases of equality. We also deduce several otherconsequences of the general subadditivity inequality, including ageneralization of Hadamard's inequality for determinants. Finally, we alsoprove a second duality theorem relating superadditivity of the Fisherinformation and a sharp convolution type inequality for the fundamentaleigenvalues of Schr\"odinger operators. Though we focus mainly on the case ofrandom variables in $\R^n$ in this paper, we discuss extensions to othersettings as well.

Subadditivity of the entropy and its relation to Brascamp-Lieb type inequalities