Stability of a reverse isoperimetric inequality

In this note we will present a stability property of the reverse isoperimetric inequality newly obtained in [S.L. Pan, H. Zhang, A reverse isoperimetric inequality for convex plane curves, Beiträge Algebra Geom. 48 (2007) 303–308], which states that if K is a convex domain in the plane with perimeter p ( K ) and area a ( K ) , then one gets p ( K ) 2 ⩽ 4 π ( a ( K ) + | a ˜ ( K ) | ) , where a ˜ ( K ) denotes the oriented area of the domain enclosed by the locus of curvature centers of the boundary curve ∂K, and the equality holds if and only if K is a circular disc.