Analyzing Convex Plane Curves with Support Function

The support function provides an intuitive framework for analyzing convex plane curves, offering insights into properties such as width, while being analogous to the curve itself. Traditionally, it has been applied to constant width curves and classical results like the Blaschke-Lebesgue theorem. In this paper, we extend the use of the support function, its derivatives, and its Fourier series to derive basic formulae for convex plane curves, proving both the isoperimetric inequality and a reverse isoperimetric inequality regarding the area of the curve’s evolute. Additionally, we extend our investigation beyond constant width curves to two broader categories: curves with a known, though not necessarily constant, width function, and curves with a bounded width function. For these generalized curves, we derive novel bounds for their perimeter, an upper bound for their area, and construct the equality cases. Furthermore, we introduce and prove a stricter isoperimetric inequality for bounded width curves.