10,657 research outputs found
On -point configuration sets with nonempty interior
We give conditions for -point configuration sets of thin sets to have
nonempty interior, applicable to a wide variety of configurations. This is a
continuation of our earlier work \cite{GIT19} on 2-point configurations,
extending a theorem of Mattila and Sj\"olin \cite{MS99} for distance sets in
Euclidean spaces. We show that for a general class of -point configurations,
the configuration set of a -tuple of sets, , has
nonempty interior provided that the sum of their Hausdorff dimensions satisfies
a lower bound, dictated by optimizing -Sobolev estimates of associated
generalized Radon transforms over all nontrivial partitions of the points
into two subsets. We illustrate the general theorems with numerous specific
examples. Applications to 3-point configurations include areas of triangles in
or the radii of their circumscribing circles; volumes of pinned
parallelepipeds in ; and ratios of pinned distances in and . Results for 4-point configurations include cross-ratios
on , triangle area pairs determined by quadrilaterals in , and dot products of differences in .Comment: 32 pages, no figure
Bylaw Governance
This article argues that Delaware corporate law permits shareholders to use bylaws to circumscribe the managerial authority of the board of directors. While shareholders cannot mandate action by the board, they can enact specific prohibitions on its behavior, so long as the board retains enough discretion to implement—in practice, not merely in theory—its managerial policies by other means. The use of such circumscribing bylaws to discourage shirking (or analogous managerial abuses) by the directors or officers resembles the use of negative covenants in debt contracts that seek to prevent the debtor from squandering assets. Bylaw governance thus subtly but significantly reallocates governance power within the corporation, so as to reduce the agency costs of management. Its legal validity should also prompt courts and scholars alike to focus less on the quantity of power wielded by the shareholders, and more on the ways that power can be configured to produce managerial efficiencies
Delaunay Edge Flips in Dense Surface Triangulations
Delaunay flip is an elegant, simple tool to convert a triangulation of a
point set to its Delaunay triangulation. The technique has been researched
extensively for full dimensional triangulations of point sets. However, an
important case of triangulations which are not full dimensional is surface
triangulations in three dimensions. In this paper we address the question of
converting a surface triangulation to a subcomplex of the Delaunay
triangulation with edge flips. We show that the surface triangulations which
closely approximate a smooth surface with uniform density can be transformed to
a Delaunay triangulation with a simple edge flip algorithm. The condition on
uniformity becomes less stringent with increasing density of the triangulation.
If the condition is dropped completely, the flip algorithm still terminates
although the output surface triangulation becomes "almost Delaunay" instead of
exactly Delaunay.Comment: This paper is prelude to "Maintaining Deforming Surface Meshes" by
Cheng-Dey in SODA 200
Radii minimal projections of polytopes and constrained optimization of symmetric polynomials
We provide a characterization of the radii minimal projections of polytopes
onto -dimensional subspaces in Euclidean space \E^n. Applied on simplices
this characterization allows to reduce the computation of an outer radius to a
computation in the circumscribing case or to the computation of an outer radius
of a lower-dimensional simplex. In the second part of the paper, we use this
characterization to determine the sequence of outer -radii of regular
simplices (which are the radii of smallest enclosing cylinders). This settles a
question which arose from the incidence that a paper by Wei{\ss}bach (1983) on
this determination was erroneous. In the proof, we first reduce the problem to
a constrained optimization problem of symmetric polynomials and then to an
optimization problem in a fixed number of variables with additional integer
constraints.Comment: Minor revisions. To appear in Advances in Geometr
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