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90 votes
5 answers
4k views

Does this property characterize straight lines in the plane?

Take a plane curve $\gamma$ and a disk of fixed radius whose center moves along $\gamma$. Suppose that $\gamma$ always cuts the disk in two simply connected regions of equal area. Is it true that $\...
1 vote
0 answers
94 views

Constant width curves and inscribed/ circumscribed ellipses

It is known (see for example the Wikipedia entry on the Reuleaux triangle) that for every curve of constant width (CCW), the largest inscribed circle and the smallest circumscribed circle are ...
4 votes
0 answers
101 views

Closed curves with minimal total curvature in the unit circle

Chakerian proved in this paper that a closed curve of length L in the unit ball in $\mathbb{R}^n$ has total curvature at least L. In this later paper Chakerian gave a simpler proof and noted that ...
16 votes
1 answer
667 views

Can a shape rolling inside itself reproduce that shape?

Q. Is the circle the only shape that, when rolling inside itself, has a point that draws out a scaled copy of itself? Let $C$ be a simple, closed, smooth curve in the plane. (Likely "smooth" can be ...
2 votes
0 answers
34 views

Least Width of Planar Unimodal Curves with Unit Diameter

I am currently trying to find a way to define some notion of "roundness" for subtours in graphs and that definition should only be based on the comparing (sums of) edge length and on the order in ...