Geodesic
Geometric Inequalities for Plane Convex Bodies
Canadian mathematical bulletin, Tome 22 (1979) no. 1, pp. 9-16

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In what follows we shall mean by a plane convex body K a compact convex subset of the Euclidean plane having nonempty interior. We shall denote by h (K, θ) the supporting function of K restricted to the unit circle. This measures the signed distances from the origin to the supporting line of K with outward normal (cos θ, sin θ). The right hand and left hand derivatives of h (K, θ) exist everywhere and are equal except on a countable set. 
Chakerian, G. D. Geometric Inequalities for Plane Convex Bodies. Canadian mathematical bulletin, Tome 22 (1979) no. 1, pp. 9-16. doi: 10.4153/CMB-1979-002-3
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