17 research outputs found
Some sharp estimates for convex hypersurfaces of pinched normal curvature
For a convex domain bounded by the hypersurface in a space
of constant curvature we give sharp bounds on the width of a spherical
shell with radii and that can enclose , provided that
normal curvatures of are pinched by two positive constants.
Furthermore, in the Euclidean case we also present sharp estimates for the
quotient . From the obtained estimates we derive stability results for
almost umbilical hypersurfaces in the constant curvature spaces.Comment: 2 figure
Inradius estimates for convex domains in 2-dimensional Alexandrov spaces
We obtain sharp lower bounds on the radii of inscribed balls for strictly
convex isoperimetric domains lying in a 2-dimensional Alexandrov metric space
of curvature bounded below. We also characterize the case when such bounds are
attained.Comment: 1 figur
A sausage body is a unique solution for a reverse isoperimetric problem
We consider the class of -concave bodies in ; that
is, convex bodies with the property that each of their boundary points supports
a tangent ball of radius that lies locally (around the boundary
point) inside the body. In this class we solve a reverse isoperimetric problem:
we show that the convex hull of two balls of radius (a sausage
body) is a unique volume minimizer among all -concave bodies of given
surface area. This is in a surprising contrast to the standard isoperimetric
problem for which, as it is well-known, the unique maximizer is a ball. We
solve the reverse isoperimetric problem by proving a reverse quermassintegral
inequality, the second main result of this paper.Comment: 1 figur
A Combinatorial classification of postcritically fixed Newton maps
We give a combinatorial classification for the class of postcritically fixed
Newton maps of polynomials as dynamical systems. This lays the foundation for
classification results of more general classes of Newton maps.
A fundamental ingredient is the proof that for every Newton map
(postcritically finite or not) every connected component of the basin of an
attracting fixed point can be connected to through a finite chain of
such components.Comment: 37 pages, 5 figures, published in Ergodic Theory and Dynamical
Systems (2018). This is the final author file before publication. Text
overlap with earlier arxiv file observed by arxiv system relates to an
earlier version that was erroneously uploaded separately. arXiv admin note:
text overlap with arXiv:math/070117
A Hyperbolic View of the Seven Circles Theorem
In this note, we will explain the connection between the Seven Circles
Theorem and hyperbolic geometry, then prove a stronger result about hyperbolic
geometry hexagons which implies the Seven Circles Theorem as a special case.Comment: 5 figure