Some notes on directional curvature of a convex body in ℝn. (2nd November 2022)

• Record Type: Journal Article
• Authors: Pereira, F. F.
• Is Part Of: Optimization. Volume 71:Number 11(2022)
• Page Start: 3313
• Page End: 3325
• Publication Date: 2022-11-02
• Subjects: Convex set -- curvature -- tangent vector -- Minkowski functional -- duality mapping; 46B20 -- 52A20 -- 53A04 -- 53A05; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2022.2052289
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract : Take a point ξ on the boundary of a convex body F in 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 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.