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Entropy bounds on abelian groups and the ruzsa divergence
Over the past few years, a family of interesting new inequalities for the
entropies of sums and differences of random variables has been developed by
Ruzsa, Tao and others, motivated by analogous results in additive
combinatorics. The present work extends these earlier results to the case of
random variables taking values in or, more generally, in
arbitrary locally compact and Polish abelian groups. We isolate and study a key
quantity, the Ruzsa divergence between two probability distributions, and we
show that its properties can be used to extend the earlier inequalities to the
present general setting. The new results established include several variations
on the theme that the entropies of the sum and the difference of two
independent random variables severely constrain each other. Although the
setting is quite general, the result are already of interest (and new) for
random vectors in . In that special case, quantitative bounds are
provided for the stability of the equality conditions in the entropy power
inequality; a reverse entropy power inequality for log-concave random vectors
is proved; an information-theoretic analog of the Rogers-Shephard inequality
for convex bodies is established; and it is observed that some of these results
lead to new inequalities for the determinants of positive-definite matrices.
Moreover, by considering the multiplicative subgroups of the complex plane, one
obtains new inequalities for the differential entropies of products and ratios
of nonzero, complex-valued random variables
Two remarks on generalized entropy power inequalities
This note contributes to the understanding of generalized entropy power
inequalities. Our main goal is to construct a counter-example regarding
monotonicity and entropy comparison of weighted sums of independent identically
distributed log-concave random variables. We also present a complex analogue of
a recent dependent entropy power inequality of Hao and Jog, and give a very
simple proof.Comment: arXiv:1811.00345 is split into 2 papers, with this being on
Improved bounds for Hadwiger's covering problem via thin-shell estimates
A central problem in discrete geometry, known as Hadwiger's covering problem,
asks what the smallest natural number is such that every
convex body in can be covered by a union of the interiors of
at most of its translates. Despite continuous efforts, the
best general upper bound known for this number remains as it was more than
sixty years ago, of the order of .
In this note, we improve this bound by a sub-exponential factor. That is, we
prove a bound of the order of for some universal
constant .
Our approach combines ideas from previous work by Artstein-Avidan and the
second named author with tools from Asymptotic Geometric Analysis. One of the
key steps is proving a new lower bound for the maximum volume of the
intersection of a convex body with a translate of ; in fact, we get the
same lower bound for the volume of the intersection of and when they
both have barycenter at the origin. To do so, we make use of measure
concentration, and in particular of thin-shell estimates for isotropic
log-concave measures.
Using the same ideas, we establish an exponentially better bound for
when restricting our attention to convex bodies that are
. By a slightly different approach, an exponential improvement is
established also for classes of convex bodies with positive modulus of
convexity
Conditional R\'enyi entropy and the relationships between R\'enyi capacities
The analogues of Arimoto's definition of conditional R\'enyi entropy and
R\'enyi mutual information are explored for abstract alphabets. These
quantities, although dependent on the reference measure, have some useful
properties similar to those known in the discrete setting. In addition to
laying out some such basic properties and the relations to R\'enyi divergences,
the relationships between the families of mutual informations defined by
Sibson, Augustin-Csisz\'ar, and Lapidoth-Pfister, as well as the corresponding
capacities, are explored.Comment: 17 pages, 1 figur