Some notes on directional curvature of a convex body in ℝ n

Abstract

Take a point ξ on the boundary of a convex body F in , near which the boundary is given by an implicit equation. We present some notes on the formula, proposed in Pereira [A directional curvature formula for convex bodies in . J Math Anal Appl. 2022;506(1):125656.], for calculating the curvature of F at ξ in the direction of its any tangent vector. Namely, we see that our formula is equivalent to the existing one for the curvature of a certain curve given by the intersection of n−1 implicit equations, but it is easier to apply. Furthermore, we show that when the directional curvature of F is positive, there is the directional derivative of the Minkowski functional of the polar set , and we propose a formula to calculate it.