Some notes on directional curvature of a convex body in ℝⁿ

Abstract

Take a point ξ on the boundary of a convex body F in ${\mathbb{R}}^n$, 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 ${\mathbb{R}}^n$. 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 $F^o$, and we propose a formula to calculate it.

Keywords:

• Convex set
• curvature
• tangent vector
• Minkowski functional
• duality mapping

AMS CLASSIFICATIONS:

• 46B20
• 52A20
• 53A04
• 53A05

Acknowledgments

The author gratefully acknowledge the Editor and the Referees for valuable comments that have contributed to the improvement of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the Center for Research in Mathematics and Applications (CIMA) related with the Differential Equations and Optimization (DEO) group, under Grant UIDB/04674/2020 of FCT-Fundação para a Ciência e a Tecnologia, Portugal.