A reverse isoperimetric inequality for embedded starshaped plane curves

Abstract

In this note, we present a reverse isoperimetric inequality for embedded starshaped closed plane curves, which states that if K is a starshaped domain with perimeter p(K) and area a(K), then one gets

$$\begin{aligned} p(K)^2 \le 4\pi \Big ((a(K)+\tilde{a}(K)\Big ), \end{aligned}$$

where $\tilde{a}(K)$ denotes the oriented area of the domain enclosed by $\beta$ (defined in Section 2), and equality holds if and only if K is a disc.