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1. CMB 2009 (vol 52 pp. 424)
Covering Discs in Minkowski Planes We investigate the following version of the circle covering
problem in strictly convex (normed or) Minkowski planes: to cover
a circle of largest possible diameter by $k$ unit circles. In
particular, we study the cases $k=3$, $k=4$, and $k=7$. For $k=3$
and $k=4$, the diameters under consideration are described in
terms of side-lengths and circumradii of certain inscribed regular
triangles or quadrangles. This yields also simple explanations of
geometric meanings that the corresponding homothety ratios have.
It turns out that basic notions from Minkowski geometry play an
essential role in our proofs, namely Minkowskian bisectors,
$d$-segments, and the monotonicity lemma.
Keywords:affine regular polygon, bisector, circle covering problem, circumradius, $d$-segment, Minkowski plane, (strictly convex) normed plane Categories:46B20, 52A21, 52C15 |
2. CMB 2006 (vol 49 pp. 321)
Polygons with Prescribed Gauss Map in Hadamard Spaces and Euclidean Buildings We show that given a stable weighted configuration on the asymptotic
boundary of a
locally compact Hadamard space, there is a polygon with Gauss
map prescribed by the given weighted configuration.
Moreover, the same result holds for
semistable configurations on arbitrary Euclidean buildings.
Keywords:Euclidean buildings, Hadamard spaces, polygons Category:53C20 |
3. CMB 2002 (vol 45 pp. 417)
On Deformations of the Complex Structure on the Moduli Space of Spatial Polygons For an integer $n \geq 3$, let $M_n$ be the moduli space of spatial polygons
with $n$ edges. We consider the case of odd $n$. Then $M_n$ is a Fano
manifold of complex dimension $n-3$. Let $\Theta_{M_n}$ be the
sheaf of germs of holomorphic sections of the tangent bundle
$TM_n$. In this paper, we prove $H^q (M_n,\Theta_{M_n})=0$ for all
$q \geq 0$ and all odd $n$. In particular, we see that the moduli
space of deformations of the complex structure on $M_n$ consists of
a point. Thus the complex structure on $M_n$ is locally rigid.
Keywords:polygon space, complex structure Categories:14D20, 32C35 |