The ultimate planar convex hull algorithm
SIAM Journal on Computing
Angle-restricted tours in the plane
Computational Geometry: Theory and Applications
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Maximizing maximal angles for plane straight-line graphs
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
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Giving a (partial) solution to a problem of Fekete [Geometry and the Traveling Salesman Problem, Ph.D. thesis, University of Waterloo, Waterloo, ON, Canada, 1992] and Fekete and Woeginger [Comput. Geom., 8 (1997), pp. 195-218], we show that given a finite set $X$ of points in the plane, it is possible to find a polygonal path with $|X|-1$ segments and with vertex set $X$ so that every angle on the polygonal path is at least $\pi/9$. According to a conjecture of Fekete and Woeginger, $\pi/9$ can be replaced by $\pi/6$. Previously, the result has not been known with any positive constant. We show further that the same result holds, with an angle smaller than $\pi/9$, in higher dimensions.