Semidefinite programming for optimizing convex bodies under width constraints

Abstract

We consider the problem of minimizing a functional (such as the area, perimeter and surface) within the class of convex bodies whose support functions are trigonometric polynomials. The convexity constraint is transformed via the Fejér–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.

Keywords:

• convexity
• optimization
• semidefinite programming
• support function

AMS Subject Classification :

• 49M27
• 90C22
• 52C05

Acknowledgements

The authors would like to thank Jean-Baptiste Hiriart-Urruty for many fruitful discussions and for being at the origin of our collaboration. The second author acknowledges support by project No. 103/10/0628 of the Grant Agency of the Czech Republic. This work was partly supported by a project PEPS of the French CNRS Institutes of Mathematics, Information Sciences and Engineering Sciences. \reversemarginpar