Abstract
AbstractIn response to an open problem raised by S. Rabinowitz, we prove that $$ \begin{align*} \begin{array} [c]{l} \left( \left( x^{2}+y^{2}\right) {}^{2}+8y\left( y^{2}-3x^{2}\right) \right) {}^{2}+432y\left( y^{2}-3x^{2}\right) \left( 351-10\left( x^{2}+y^{2}\right) \right) \\ =567^{3}+28\left( x^{2}+y^{2}\right) {}^{3}+486\left( x^{2}+y^{2}\right) \left( 67\left( x^{2}+y^{2}\right) -567\times18\right) \end{array} \end{align*} $$ is the equation of a plane convex curve of constant width.
Highlights
Rabinowitz [5] found that the zero set of the following polynomial P 2 R [X; Y ] forms a non-circular algebraic curve of constant width in R2: P (x; y) := (x2 + y2)4 45 (x2 + y2)3 +7950960 (x2 + y2) + 16 (x2 +x (x2 3y2) 16 (x2 + y2)2
Any hedgehog whose support function h : S1 ! R is of the form h ( ) = r sin (3 ), for some constant r, is a hedgehog of constant width 2r
Each of the above two examples of algebraic curves of constant width admits an axis of symmetry in R2
Summary
A disk has the property that it can be rotated between two ...xed parallel lines without losing contact with either line It has been known for a long time that there are many other plane convex bodies with the same property. A non-circular plane convex curve of constant width can be smooth, and not having any circular arc in its boundary. Rabinowitz [5] found that the zero set of the following polynomial P 2 R [X; Y ] forms a non-circular algebraic curve of constant width in R2:. It has been proved by Bardet and Bayen [1, Cor. 2.1] that the degree of P , that is 8, is the minimum possible degree for a non-circular plane convex curve of constant width. We give an example of a non-circular smooth algebraic curve of constant width whose equation is simpler than the one of Rabinowitz. We notice that we can deduce from it (relatively) simple examples in higher dimensions
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