Stability of reverse isoperimetric inequalities in the plane: area, Cheeger, and inradius

In this paper, we present sharp stability results for various reverse isoperimetric problems in $\mathbb R^2$. Specifically, we prove the stability of the reverse isoperimetric inequality for $λ$-convex bodies -- convex bodies with the property that each of their boundary points $p$ supports a ball of radius $1/λ$ so that the body lies inside the ball in a neighborhood of $p$. For convex bodies with smooth boundaries, $λ$-convexity is equivalent to having the curvature of the boundary bounded below by $λ> 0$. Additionally, within this class of convex bodies, we establish stability for the reverse inradius inequality and the reverse Cheeger inequality. Even without its stability version, the sharp reverse Cheeger inequality is new in dimension $2$.

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