702 research outputs found
Involutes of Polygons of Constant Width in Minkowski Planes
Consider a convex polygon P in the plane, and denote by U a homothetical copy
of the vector sum of P and (-P). Then the polygon U, as unit ball, induces a
norm such that, with respect to this norm, P has constant Minkowskian width. We
define notions like Minkowskian curvature, evolutes and involutes for polygons
of constant U-width, and we prove that many properties of the smooth case,
which is already completely studied, are preserved. The iteration of involutes
generates a pair of sequences of polygons of constant width with respect to the
Minkowski norm and its dual norm, respectively. We prove that these sequences
are converging to symmetric polygons with the same center, which can be
regarded as a central point of the polygon P.Comment: 20 pages, 11 figure
The Rosenthal-Szasz inequality for normed planes
We aim to study the classical Rosenthal-Szasz inequality for a plane whose
geometry is given by a norm. This inequality states that the bodies of constant
width have the largest perimeter among all planar convex bodies of given
diameter. In the case where the unit circle of the norm is given by a Radon
curve, we obtain an inequality which is completely analogous to the Euclidean
case. For arbitrary norms we obtain an upper bound for the perimeter calculated
in the anti-norm, yielding an analogous characterization of all curves of
constant width. To derive these results, we use methods from the differential
geometry of curves in normed planes
New Moduli for Banach Spaces
Modifying the moduli of supporting convexity and supporting smoothness, we
introduce new moduli for Banach spaces which occur, e.g., as lengths of catheti
of right-angled triangles (defined via so-called quasi-orthogonality). These
triangles have two boundary points of the unit ball of a Banach space as
endpoints of their hypotenuse, and their third vertex lies in a supporting
hyperplane of one of the two other vertices. Among other things it is our goal
to quantify via such triangles the local deviation of the unit sphere from its
supporting hyperplanes. We prove respective Day-Nordlander type results,
involving generalizations of the modulus of convexity and the modulus of
Bana\'{s}
Conics in normed planes
We study the generalized analogues of conics for normed planes by using the
following natural approach: It is well known that there are different metrical
definitions of conics in the Euclidean plane. We investigate how these
definitions extend to normed planes, and we show that in this more general
framework these different definitions yield, in almost all cases, different
classes of curves.Comment: 12 pages, 8 figure
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