Positive Center Sets of Convex Curves

Abstract

In this paper we shall investigate the positive center set $${\mathfrak {P}}(\gamma )$$P(?) of a convex curve $$\gamma $$? and show that $${\mathfrak {P}}(\gamma )$$P(?) has only one point if and only if $$\gamma $$? is a circle; $${\mathfrak {P}}(\gamma )$$P(?) is a segment if and only if $$\gamma $$? is a sausage curve; if $$\gamma $$? is a strictly convex non-circular curve, then $${\mathfrak {P}}(\gamma )$$P(?) is a domain of positive area; and furthermore, if $$\gamma $$? is a constant width curve, then $${\mathfrak {P}}(\gamma )$$P(?) is its inner parallel body $$K_{-r_1}$$K-r1.