JP Journal of Geometry and Topology
Volume 1, Issue 1, Pages 1 - 36
(March 2001)
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EQUIDISTANT THICKNESS IN A SPACE OF CONSTANT CURVATURE
B. V. Dekster (Canada)
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Abstract:
First, we discuss here
two existing (a la Santaló) definitions of minimum
width of a convex body in an n-dimensional
space Xn
of constant curvature. These definitions are based on
placing the body in a smallest part of Xn
bounded by a pair of hyperplanes. They have some
shortcomings. For instance, in the hyperbolic plane
there exists a convex body of arbitrarily small
minimum width that contains an arbitrary large circle.
We come up with another definition of the minimum
width called equidistant thickness of a compact
set. It is based on placing the set in a smallest part
of Xn
between two hypersurfaces equidistant to a
hyperplane.
This preserves some natural properties of minimum
width. Then we apply this new definition to estimate
how thick at least a simplex in Xn
should be when its edge lengths fall within a
prescribed range. |
| Keywords and phrases: minimum width, spherical and hyperbolic geometries, equidistant thickness, simplex. |
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