Hostname: page-component-f554764f5-rj9fg Total loading time: 0 Render date: 2025-04-21T10:23:48.887Z Has data issue: false hasContentIssue false
  • English
  • Français

Convexity of the radial sum of a star body and a ball

Part of: General convexity Approximations and expansions

Published online by Cambridge University Press:  04 September 2023

Shigehiro Sakata Open the ORCID record for Shigehiro Sakata [Opens in a new window]
Shigehiro Sakata*
Affiliation:
Department of Applied Mathematics, Fukuoka University, 8-19-1 Nanakuma, Jonan, Fukuoka 814-0180, Japan
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the convexity of the radial sum of two convex bodies containing the origin. Generally, the radial sum of two convex bodies containing the origin is not convex. We show that the radial sum of a star body (with respect to the origin) and any large centered ball is convex, which produces a pair of convex bodies containing the origin whose radial sum is convex.

We also investigate the convexity of the intersection body of a convex body containing the origin. Generally, the intersection body of a convex body containing the origin is not convex. Busemann’s theorem states that the intersection body of any centered convex body is convex. We are interested in how to construct convex intersection bodies from convex bodies without any symmetry (especially, central symmetry). We show that the intersection body of the radial sum of a star body (with respect to the origin) and any large centered ball is convex, which produces a convex body with no symmetries whose intersection body is convex.

Keywords

Convexity of the radial sum of convex bodiesBusemann’s theoremconvexity of intersection bodies

MSC classification

Primary: 52A39: Mixed volumes and related topics 52A40: Inequalities and extremum problems
Secondary: 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 2 , June 2024 , pp. 289 - 303
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

The author is partially supported by JSPS KAKENHI (Grant No. 17K14191) and JSPS Overseas Research Fellowships (Grant No. 201860263).

References

Alfonseca, M. A. and Kim, J., On the local convexity of intersection bodies of revolution . Can. J. Math. 67(2015), no. 1, 3–27.CrossRefGoogle Scholar
Busemann, H., A theorem of convex bodies of the Brunn–Minkowski type . Proc. Natl. Acad. Sci. USA 35(1949), 27–31.CrossRefGoogle ScholarPubMed
Busemann, H. and Petty, C. M., Problems on convex bodies . Math. Scand. 4(1956), 88–94.CrossRefGoogle Scholar
Gardner, R. J., Intersection bodies and the Busemann–Petty problem . Trans. Amer. Math. Soc. 342(1994), no. 1, 435–445.CrossRefGoogle Scholar
Gardner, R. J., A positive answer to the Busemann–Petty problem in three dimensions . Ann. Math. 140(1994), no. 2, 435–447.CrossRefGoogle Scholar
Gardner, R. J., Geometric tomography. 2nd ed., Encyclopedia of Mathematics and Its Applications, 58, Cambridge University Press, Cambridge, 2006.CrossRefGoogle Scholar
Gardner, R. J., Koldobsky, A., and Schlumprecht, T., An analytic solution to the Busemann–Petty problem on sections of convex bodies . Ann. Math. 149(1999), no. 2, 691–703.CrossRefGoogle Scholar
Hansen, G., Herburt, I., Martini, H., and Moszyńska, M., Starshaped sets . Aequat. Math. 94(2020), 1001–1092.CrossRefGoogle Scholar
Kim, J., Yaskin, V., and Zvavitch, A., The geometry of $p$ -convex intersection bodies. Adv. Math. 226(2011), no. 6, 5320–5337.CrossRefGoogle Scholar
Lutwak, E., Intersection bodies and dual mixed volumes . Adv. Math. 71(1988), 232–261.CrossRefGoogle Scholar
Meyer, M. and Reisner, S., The convex intersection body of a convex body . Glasg. Math. J. 53(2011), 523–534.CrossRefGoogle Scholar
Moszyńska, M., Selected topics in convex geometry, Birkhäuser, Boston, 2006.Google Scholar
Zhang, G. Y., Centered bodies and dual mixed volumes . Trans. Amer. Math. Soc. 345(1994), no. 2, 777–801.CrossRefGoogle Scholar
Zhang, G. Y., A positive solution to the Busemann–Petty problem in ℝ4. Ann. Math. 149(1999), no. 2, 535–543.CrossRefGoogle Scholar