113 research outputs found
Convex Bodies of Constant Width and Constant Brightness
In 1926 S. Nakajima (= A. Matsumura) showed that any convex body in
with constant width, constant brightness, and boundary of class is a
ball. We show that the regularity assumption on the boundary is unnecessary, so
that balls are the only convex bodies of constant width and brightness.Comment: 20 page
Entropy on Spin Factors
Recently it has been demonstrated that the Shannon entropy or the von Neuman
entropy are the only entropy functions that generate a local Bregman
divergences as long as the state space has rank 3 or higher. In this paper we
will study the properties of Bregman divergences for convex bodies of rank 2.
The two most important convex bodies of rank 2 can be identified with the bit
and the qubit. We demonstrate that if a convex body of rank 2 has a Bregman
divergence that satisfies sufficiency then the convex body is spectral and if
the Bregman divergence is monotone then the convex body has the shape of a
ball. A ball can be represented as the state space of a spin factor, which is
the most simple type of Jordan algebra. We also study the existence of recovery
maps for Bregman divergences on spin factors. In general the convex bodies of
rank 2 appear as faces of state spaces of higher rank. Therefore our results
give strong restrictions on which convex bodies could be the state space of a
physical system with a well-behaved entropy function.Comment: 30 pages, 6 figure
Introducing symplectic billiards
In this article we introduce a simple dynamical system called symplectic
billiards. As opposed to usual/Birkhoff billiards, where length is the
generating function, for symplectic billiards symplectic area is the generating
function. We explore basic properties and exhibit several similarities, but
also differences of symplectic billiards to Birkhoff billiards.Comment: 41 pages, 16 figure
LR characterization of chirotopes of finite planar families of pairwise disjoint convex bodies
We extend the classical LR characterization of chirotopes of finite planar
families of points to chirotopes of finite planar families of pairwise disjoint
convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a
chirotope of finite planar families of pairwise disjoint convex bodies if and
only if for every 3-, 4-, and 5-subset J of I the restriction of \c{hi} to the
set of 3-subsets of J is a chirotope of finite planar families of pairwise
disjoint convex bodies. Our main tool is the polarity map, i.e., the map that
assigns to a convex body the set of lines missing its interior, from which we
derive the key notion of arrangements of double pseudolines, introduced for the
first time in this paper.Comment: 100 pages, 73 figures; accepted manuscript versio
Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional II: Smooth case
The paper is centered around a new proof of the infinitesimal rigidity of
smooth closed surfaces with everywhere positive Gauss curvature. We use a
reformulation that replaces deformation of an embedding by deformation of the
metric inside the body bounded by the surface. The proof is obtained by
studying derivatives of the Hilbert-Einstein functional with boundary term.
This approach is in a sense dual to proving the Gauss infinitesimal rigidity,
that is rigidity with respect to the Gauss curvature parametrized by the Gauss
map, by studying derivatives of the volume bounded by the surface. We recall
that Blaschke's classical proof of the infinitesimal rigidity is also related
to the Gauss infinitesimal rigidity, but in a different way: while Blaschke
uses Gauss rigidity of the same surface, we use the Gauss rigidity of the polar
dual. In the spherical and in the hyperbolic-de Sitter space, there is a
perfect duality between the Hilbert-Einstein functional and the volume, as well
as between both kinds of rigidity. We also indicate directions for future
research, including the infinitesimal rigidity of convex cores of hyperbolic
3--manifolds.Comment: 60 page
Recommended from our members
Mini-Workshop: Valuations and Integral Geometry
As a generalization of the notion of measure, valuations have long played a central role in the integral geometry of convex sets. In recent years there has been a series of striking developments. Several examples were presented at this meeting, e.g. the work of Bernig and Fu on the integral geometry of groups acting transitively on the unit sphere, that of Hug and Schneider on kinematic and Crofton formulas for tensor valued valuations and a series of results by Ludwig and Reitzner on classifications of affine invariant notions of surface areas and of convex body valued valuations
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