This paper deals with a new curvature flow for closed convex plane curves which shortens the length of the evolving curve but expands the area it bounds and makes the evolving curve more and more circular during the evolution process. And the final shape of the evolving curve will be a circle (as the time t goes to infinity). This flow is determined by a coupled system concerning both local and global geometric quantities of the evolving curve.

No abstract

Abstract. It is proved that if γ is a closed strictly convex curve in the plane with length L and area A , then L 2 4πA + 2π|Ã|, with equality holding if and only if γ is a circle, whereà denotes the oriented area enclosed by the locus of curvature centers of γ .Mathematics subject classification (2010): 52A38, 52A40.

We introduce a function w k (θ ) for closed convex plane curves, and then prove a geometric inequality involving w k (θ ) and the area A enclosed by the curve. As a by-product, we give a new proof of the classical isoperimetric inequality. Finally, we give some properties of convex curves with w k (θ ) being constant and pose an open problem motivated by the elegant Blaschke-Lebesgue theorem.

Motivated by Gage [On an area-preserving evolution equation for plane curves, in Nonlinear Problems in Geometry, ed. D. M. DeTurck, Contemporary Mathematics, Vol. 51 (American Mathematical Society, Providence, RI, 1986), pp. 51–62] and Ma–Cheng [A non-local area preserving curve flow, preprint (2009), arXiv:0907.1430v2, [math.DG]], in this paper, an area-preserving flow for convex plane curves is presented. This flow will decrease the perimeter of the evolving curve and make the curve more and more circular during the evolution process. And finally, as t goes to infinity, the limiting curve will be a finite circle in the C∞ metric.