5 research outputs found
On the degree of the polynomial defining a planar algebraic curves of constant width
In this paper, we consider a family of closed planar algebraic curves
which are given in parametrization form via a trigonometric
polynomial . When is the boundary of a compact convex set, the
polynomial represents the support function of this set. Our aim is to
examine properties of the degree of the defining polynomial of this family of
curves in terms of the degree of . Thanks to the theory of elimination, we
compute the total degree and the partial degrees of this polynomial, and we
solve in addition a question raised by Rabinowitz in \cite{Rabi} on the lowest
degree polynomial whose graph is a non-circular curve of constant width.
Computations of partial degrees of the defining polynomial of algebraic
surfaces of constant width are also provided in the same way.Comment: 13 page
Semidefinite programming for optimizing convex bodies under width constraints
International audienceWe consider the problem of minimizing a functional (like the area, perimeter, surface) within the class of convex bodies whose support functions are trigonometric polynomials. The convexity constraint is transformed via the Fejer-Riesz theorem on positive trigonometric polynomials into a semidefinite programming problem. Several problems such as the minimization of the area in the class of constant width planar bodies, rotors and space bodies of revolution are revisited. The approach seems promising to investigate more difficult optimization problems in the class of three-dimensional convex bodies