Let $f(x,y)=0$ be a (smooth) simple closed curve $C$ on the plane and $R$ the region bounded by $C$ (appropriately oriented). Assume the origin lies in the interior of $R$.

QUESTION. Let $r=\sqrt{x^2+y^2}$. Is this true? $$\int_Cr\,ds\geq 2\cdot Area(R).$$ Equality iff $C$ is a circle.

Let $f(x,y)=0$ be a (smooth) simple closed curve $C$ on the plane and $R$ the region bounded by $C$ (appropriately oriented). Assume the origin lies in the interior of $R$.

QUESTION. Let $r=\sqrt{x^2+y^2}$. Is this true? $$\int_Cr\,ds\geq 2\cdot Area(R).$$ Equality iff $C$ is a circle centered at the origin.