41 research outputs found
A Lower Estimate for the Modified Steiner Functional
We prove inequality (1) for the modified Steiner functional A(M), which
extends the notion of the integral of mean curvature for convex surfaces.We
also establish an exression for A(M) in terms of an integral over all
hyperplanes intersecting the polyhedralral surface M.Comment: 6 pages, Late
Illuminating spindle convex bodies and minimizing the volume of spherical sets of constant width
A subset of the d-dimensional Euclidean space having nonempty interior is
called a spindle convex body if it is the intersection of (finitely or
infinitely many) congruent d-dimensional closed balls. The spindle convex body
is called a "fat" one, if it contains the centers of its generating balls. The
core part of this paper is an extension of Schramm's theorem and its proof on
illuminating convex bodies of constant width to the family of "fat" spindle
convex bodies.Comment: 17 page
Integral geometry of complex space forms
We show how Alesker's theory of valuations on manifolds gives rise to an
algebraic picture of the integral geometry of any Riemannian isotropic space.
We then apply this method to give a thorough account of the integral geometry
of the complex space forms, i.e. complex projective space, complex hyperbolic
space and complex euclidean space. In particular, we compute the family of
kinematic formulas for invariant valuations and invariant curvature measures in
these spaces. In addition to new and more efficient framings of the tube
formulas of Gray and the kinematic formulas of Shifrin, this approach yields a
new formula expressing the volumes of the tubes about a totally real
submanifold in terms of its intrinsic Riemannian structure. We also show by
direct calculation that the Lipschitz-Killing valuations stabilize the subspace
of invariant angular curvature measures, suggesting the possibility that a
similar phenomenon holds for all Riemannian manifolds. We conclude with a
number of open questions and conjectures.Comment: 68 pages; minor change
Random line tessellations of the plane: statistical properties of many-sided cells
We consider a family of random line tessellations of the Euclidean plane
introduced in a much more formal context by Hug and Schneider [Geom. Funct.
Anal. 17, 156 (2007)] and described by a parameter \alpha\geq 1. For \alpha=1
the zero-cell (that is, the cell containing the origin) coincides with the
Crofton cell of a Poisson line tessellation, and for \alpha=2 it coincides with
the typical Poisson-Voronoi cell. Let p_n(\alpha) be the probability for the
zero-cell to have n sides. By the methods of statistical mechanics we construct
the asymptotic expansion of \log p_n(\alpha) up to terms that vanish as
n\to\infty. In the large-n limit the cell is shown to become circular. The
circle is centered at the origin when \alpha>1, but gets delocalized for the
Crofton cell, \alpha=1, which is a singular point of the parameter range. The
large-n expansion of \log p_n(1) is therefore different from that of the
general case and we show how to carry it out. As a corollary we obtain the
analogous expansion for the {\it typical} n-sided cell of a Poisson line
tessellation.Comment: 26 pages, 3 figure