On the addition of convex sets in the hyperbolic plane |
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Authors: | Kurt Leichtweiß |
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Institution: | (1) Mathematisches Institut B, Universität Stuttgart, 70550 Stuttgart, Germany |
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Abstract: | Analogue to the definition $K + L := \bigcup_{x\in K}(x + L)$ of the
Minkowski addition in the euclidean geometry it is proposed to define the
(noncommutative) addition $K \vdash L := \bigcup_{0\, \leqsl\, \rho\,\leqsl\,
a(\varphi),0\,\leqsl\,\varphi\,<\, 2\pi}T_{\rho}^{(\varphi)}(L)$ for compact,
convex and smoothly bounded sets K and
L in the hyperbolic plane $\Omega$
(Kleins model). Here $\rho = a(\varphi)$ is the representation of the boundary
$\partial$ K in geodesic polar coordinates
and $T_{\rho}^{(\varphi)}$ is the hyperbolic translation of $\Omega$ of length
$\rho$ along the line through the origin o of
direction $\varphi$. In general this addition does not preserve
convexity but nevertheless we may prove as main results: (1) $o \in$ int
$K, o \in$ int L and K,L horocyclic convex imply the strict
convexity of $K \vdash L$, and (2) in this case there exists a hyperbolic mixed
volume $V_h(K,L)$ of K and L which has a representation by a suitable
integral over the unit circle. |
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Keywords: | 52A55 51M15 |
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