title: Bonnesen's inequality
text: # Bonnesen's inequality
  
  Bonnesen's inequality is an inequality relating the length, the area, the radius of the incircle and the radius of the circumcircle of a Jordan curve. It is a strengthening of the classical isoperimetric inequality.
  
  More precisely, consider a planar simple closed curve of length $L$ bounding a domain of area $A$. Let $r$ and $R$ denote the radii of the incircle and the circumcircle. Bonnesen proved the inequality
  
  $\pi ^{2}(R-r)^{2}\leq L^{2}-4\pi A.$
  
  The term $L^{2}-4\pi A$ in the right hand side is known as the isoperimetric defect.
  
  Loewner's torus inequality with isosystolic defect is a systolic analogue of Bonnesen's inequality.
  
  