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Show that a star-shaped open subset of the plane is connected.
Solution
Every star domain, and only a star domain, can be 'shrinked into itself', i.e.: For every dilation
ratio r<1, the star domain can be dilated by a ratio r such that the dilated star domain is contained
in the original star domain.
Convex polygons are star shaped, and a convex polygon coincides with its own kernel.
Regular star polygons are star-shaped, with their center always in the kernel.
Antiparallelograms and self-intersecting Lemoine hexagons are star-shaped, with the kernel
consisting of a single point.
Visibility polygons are star-shaped as every point within them must be visible to the center by
definition.
These establish connectedness of a open subset of star shape

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Show that a star-shaped open subset of the plane is connected.So.pdf

  • 1. Show that a star-shaped open subset of the plane is connected. Solution Every star domain, and only a star domain, can be 'shrinked into itself', i.e.: For every dilation ratio r<1, the star domain can be dilated by a ratio r such that the dilated star domain is contained in the original star domain. Convex polygons are star shaped, and a convex polygon coincides with its own kernel. Regular star polygons are star-shaped, with their center always in the kernel. Antiparallelograms and self-intersecting Lemoine hexagons are star-shaped, with the kernel consisting of a single point. Visibility polygons are star-shaped as every point within them must be visible to the center by definition. These establish connectedness of a open subset of star shape