On a non-local perimeter-preserving curve evolution problem for convex plane curves

Blow-up rates for the general curve shortening flow

“…It is an interesting problem to investigate the global behavior of perimeter-preserving curve flow [9]. But we will not cover it in present paper.…”

Evolving convex curves to constant-width ones by a perimeter-preserving flow

“…Such problems arise in many fields, such as image processing [Cao 2003], phase transitions [Gurtin 1993], etc. In fact, the above evolution problem has been studied extensively, for example, for the popular curve-shortening flow [Gage 1984;Gage and Hamilton 1986;Grayson 1987], the area-preserving flows [Gage 1986;Mao et al 2013;Ma and Cheng 2014], the perimeter-preserving flows [Pan and Yang 2008;Ma and Zhu 2012] and in other related research [Angenent 1991;Chow and Tsai 1996;Andrews 1998;Urbas 1999;Chao et al 2013]. One can find more background material in the book [Chou and Zhu 2001].…”

A note on the isoperimetric inequality and its stability

“…where ρ is the radius of curvature and θ is the angle between x -axis and the outward normal vector at the corresponding point p. Moreover the equalities in (2) and (3) hold if and only if γ is a circle. It is obvious that if γ is a circle, then the locus of its curvature centers is only a point, and thus its areaà = 0.…”

“…In section 2, we recall some basic facts about plane convex geometry. In section 3, we provide a simpler proof of Theorem 1.2 by using Fourier series, which is different from the approach in [1] and [2]. In section 4, we investigate stability properties of inequality (5) (near equality implies curve nearly circular).…”