67 research outputs found
Some functional forms of Blaschke-Santal\'o inequality
We establish new functional versions of the Blaschke-Santal\'o inequality on
the volume product of a convex body which generalize to the non-symmetric
setting an inequality of K. Ball and we give a simple proof of the case of
equality. As a corollary, we get some inequalities for -concave functions
and Legendre transforms which extend the recent result of Artstein, Klartag and
Milman, with its equality case.Comment: 19 pages, to appear in Mathematische Zeitschrif
On the volume of sections of a convex body by cones
Let be a convex body in . We prove that in small
codimensions, the sections of a convex body through the centroid are quite
symmetric with respect to volume. As a consequence of our estimates we give a
positive answer to a problem posed by M. Meyer and S. Reisner regarding convex
intersection bodies.Comment: 13 page
Concentration inequalities for -concave measures of dilations of Borel sets and applications
We prove a sharp inequality conjectured by Bobkov on the measure of dilations
of Borel sets in by a -concave probability. Our result gives
a common generalization of an inequality of Nazarov, Sodin and Volberg and a
concentration inequality of Gu\'edon. Applying our inequality to the level sets
of functions satisfying a Remez type inequality, we deduce, as it is classical,
that these functions enjoy dimension free distribution inequalities and
Kahane-Khintchine type inequalities with positive and negative exponent, with
respect to an arbitrary -concave probability.Comment: 22 pages, submitte
Thin-shell concentration for convex measures
We prove that for , -concave measures on satisfy a
thin shell concentration similar to the log-concave one. It leads to a
Berry-Esseen type estimate for their one dimensional marginal distributions. We
also establish sharp reverse H\"older inequalities for -concave measures
The convexification effect of Minkowski summation
Let us define for a compact set the sequence It was independently proved by Shapley, Folkman and Starr (1969)
and by Emerson and Greenleaf (1969) that approaches the convex hull of
in the Hausdorff distance induced by the Euclidean norm as goes to
. We explore in this survey how exactly approaches the convex
hull of , and more generally, how a Minkowski sum of possibly different
compact sets approaches convexity, as measured by various indices of
non-convexity. The non-convexity indices considered include the Hausdorff
distance induced by any norm on , the volume deficit (the
difference of volumes), a non-convexity index introduced by Schneider (1975),
and the effective standard deviation or inner radius. After first clarifying
the interrelationships between these various indices of non-convexity, which
were previously either unknown or scattered in the literature, we show that the
volume deficit of does not monotonically decrease to 0 in dimension 12
or above, thus falsifying a conjecture of Bobkov et al. (2011), even though
their conjecture is proved to be true in dimension 1 and for certain sets
with special structure. On the other hand, Schneider's index possesses a strong
monotonicity property along the sequence , and both the Hausdorff
distance and effective standard deviation are eventually monotone (once
exceeds ). Along the way, we obtain new inequalities for the volume of the
Minkowski sum of compact sets, falsify a conjecture of Dyn and Farkhi (2004),
demonstrate applications of our results to combinatorial discrepancy theory,
and suggest some questions worthy of further investigation.Comment: 60 pages, 7 figures. v2: Title changed. v3: Added Section 7.2
resolving Dyn-Farkhi conjectur
Volume of the polar of random sets and shadow systems
We obtain optimal inequalities for the volume of the polar of random sets,
generated for instance by the convex hull of independent random vectors in
Euclidean space. Extremizers are given by random vectors uniformly distributed
in Euclidean balls. This provides a random extension of the Blaschke-Santalo
inequality which, in turn, can be derived by the law of large numbers. The
method involves generalized shadow systems, their connection to Busemann type
inequalities, and how they interact with functional rearrangement inequalities
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