2,305 research outputs found
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
Lorentzian area measures and the Christoffel problem
We introduce a particular class of unbounded closed convex sets of
, called F-convex sets (F stands for future). To define them, we use
the Minkowski bilinear form of signature instead of the usual
scalar product, and we ask the Gauss map to be a surjection onto the hyperbolic
space \H^d. Important examples are embeddings of the universal cover of
so-called globally hyperbolic maximal flat Lorentzian manifolds.
Basic tools are first derived, similarly to the classical study of convex
bodies. For example, F-convex sets are determined by their support function,
which is defined on \H^d. Then the area measures of order , are defined. As in the convex bodies case, they are the coefficients of the
polynomial in which is the volume of an approximation of
the convex set. Here the area measures are defined with respect to the
Lorentzian structure.
Then we focus on the area measure of order one. Finding necessary and
sufficient conditions for a measure (here on \H^d) to be the first area
measure of a F-convex set is the Christoffel Problem. We derive many results
about this problem. If we restrict to "Fuchsian" F-convex set (those who are
invariant under linear isometries acting cocompactly on \H^d), then the
problem is totally solved, analogously to the case of convex bodies. In this
case the measure can be given on a compact hyperbolic manifold.
Particular attention is given on the smooth and polyhedral cases. In those
cases, the Christoffel problem is equivalent to prescribing the mean radius of
curvature and the edge lengths respectively
Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity
In the modeling of dislocations one is lead naturally to energies
concentrated on lines, where the integrand depends on the orientation and on
the Burgers vector of the dislocation, which belongs to a discrete lattice. The
dislocations may be identified with divergence-free matrix-valued measures
supported on curves or with 1-currents with multiplicity in a lattice. In this
paper we develop the theory of relaxation for these energies and provide one
physically motivated example in which the relaxation for some Burgers vectors
is nontrivial and can be determined explicitly. From a technical viewpoint the
key ingredients are an approximation and a structure theorem for 1-currents
with multiplicity in a lattice
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