Crystals, gauges, podals, and polars

Crystals, gauges, podals, and polars are constructed for point sets and convex polygonal regions in the plane. A family of geometric extremal problems considers minimizing the area and length of the associated class of figures. Constructive procedures are described in the formation of the geometric structures and the extremal problems are solved using enumerative search techniques. Numerical examples are provided throughout to illustrate the constructions and solution methodology.