Most Reinhardt polygons are sporadic

Sporadic Reinhardt Polygons

Let n be a positive integer, not a power of two. A Reinhardt polygon is a convex n-gon that is optimal in three different geometric optimization problems: it has maximal perimeter relative to its diameter, maximal width relative to its diameter, and maximal width relative to its perimeter. For almost all n, there are many Reinhardt polygons with n sides, and many of them exhibit a particular pe...

Regular polygons are most tolerant

Convex Polygons are Self-Coverable

We introduce a new notion for geometric families called self-coverability and show that homothets of convex polygons are self-coverable. As a corollary, we obtain several results about coloring point sets such that any member of the family with many points contains all colors. This is dual (and in some cases equivalent) to the much investigated cover-decomposability problem.

A Census of Convex Lattice Polygons with at most one

We find all convex lattice polygons in the plane (up to equivalence) with at most one interior lattice point. A lattice point in the plane is a point with integer coordinates. A lattice segment is a line segment whose endpoints are lattice points. A lattice polygon is a simple polygon whose vertices are lattice points. In 1980, Arkinstall [1] proved that, up to lattice equivalence (defined belo...

Are Most-Viewed News Articles Most-Shared?

Despite many users get timely information through various social media platforms, news websites remain important mainstream media for high-quality news articles and comprehensive news coverage. Moreover, news websites are becoming well connected with the social media platform by enabling one-click sharing, allowing readers to comment on the articles, and pushing news update to social media thro...