Asymptotic behavior of the isoperimetric deficit for expanding convex plane curves

We consider the expansion of a convex closed plane curve C₀ along its outward normal direction with speed G(1/k), where k is the curvature and (Formula presented.) is a strictly increasing function. We show that if (Formula presented.), then the isoperimetric deficit (Formula presented.) of the flow converges to zero. On the other hand, if (Formula presented.), then for any number d ≥ 0 and (Formula presented.), one can choose an initial curve C₀ so that its isoperimetric deficit D(t) satisfies (Formula presented.) for all t ∈[0, ∞). Hence, without rescaling, the expanding curve C_t will not become circular. It is close to some expanding curve P_t, where each P_t is parallel to P₀. The asymptotic speed of P_t is given by the constant λ.