Journal of Mathematical Physics, Analysis, Geometry 2019, vol. 15, No 4, pp. 502-509   https://doi.org/10.15407/mag15.04.502     ( to contents , go back )

### On the Sharpness of One Integral Inequality for Closed Curves in ℝ4

Vasyl Gorkavyy

B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
E-mail: gorkaviy@ilt.kharkov.ua

Raisa Posylaieva

Kharkiv National University of Civil Engineering and Architecture, 40 Sumska Str., Kharkiv, 61002, Ukraine
E-mail: posylaevaraisa@gmail.com

Received November 29, 2018, revised January 10, 2019.

Abstract

The sharpness of the integral inequality $\int_\gamma\sqrt{k_1^2+k_2^2+k_3^2}\,ds>2\pi$ for closed curves with nowhere vanishing curvatures in $\mathbb R^4$ is discussed. We prove that an arbitrary closed curve of constant positive curvatures in $\mathbb R^4$ satisfies the inequality $\int_\gamma\sqrt{k_1^2+k_2^2+k_3^2}\,ds\geq 2\sqrt{5}\pi$.

Mathematics Subject Classification 2000: 53A04, 53A07.
Key words: closed curve, curvature, curves of constant curvatures.