105 research outputs found
Convex Geometry and its Applications
The geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDE, linear programming and, increasingly, in the study of other algorithms in computer science. High-dimensional geometry is an extremely active area of research: the participation of a considerable number of talented young mathematicians at this meeting is testament to the fact that the field is flourishing
From Steiner Formulas for Cones to Concentration of Intrinsic Volumes
The intrinsic volumes of a convex cone are geometric functionals that return
basic structural information about the cone. Recent research has demonstrated
that conic intrinsic volumes are valuable for understanding the behavior of
random convex optimization problems. This paper develops a systematic technique
for studying conic intrinsic volumes using methods from probability. At the
heart of this approach is a general Steiner formula for cones. This result
converts questions about the intrinsic volumes into questions about the
projection of a Gaussian random vector onto the cone, which can then be
resolved using tools from Gaussian analysis. The approach leads to new
identities and bounds for the intrinsic volumes of a cone, including a
near-optimal concentration inequality.Comment: This version corrects errors in Propositions 3.3 and 3.4 and in Lemma
8.3 that appear in the published versio
Convex Geometry and its Applications
The geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDE, linear programming and, increasingly, in the study of other algorithms in computer science. High-dimensional geometry, both the discrete and convex branches of it, has experienced a striking series of developments in the past 10 years. Several examples were presented at this meeting, for example the work of Rudelson et al. on conjunction matrices and their relation to confidential data analysis, that of Litvak et al. on remote sensing and a series of results by Nazarov and Ryabogin et al. on Mahler’s conjecture for the volume product of domains and their polars
Entropies from coarse-graining: convex polytopes vs. ellipsoids
We examine the Boltzmann/Gibbs/Shannon and the
non-additive Havrda-Charv\'{a}t / Dar\'{o}czy/Cressie-Read/Tsallis \
\ and the Kaniadakis -entropy \ \
from the viewpoint of coarse-graining, symplectic capacities and convexity. We
argue that the functional form of such entropies can be ascribed to a
discordance in phase-space coarse-graining between two generally different
approaches: the Euclidean/Riemannian metric one that reflects independence and
picks cubes as the fundamental cells and the symplectic/canonical one that
picks spheres/ellipsoids for this role. Our discussion is motivated by and
confined to the behaviour of Hamiltonian systems of many degrees of freedom. We
see that Dvoretzky's theorem provides asymptotic estimates for the minimal
dimension beyond which these two approaches are close to each other. We state
and speculate about the role that dualities may play in this viewpoint.Comment: 63 pages. No figures. Standard LaTe
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