Journal of Mathematical Physics, Analysis, Geometry
2019, vol. 15, No 4, pp. 502-509     ( 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

Raisa Posylaieva

Kharkiv National University of Civil Engineering and Architecture, 40 Sumska Str., Kharkiv, 61002, Ukraine

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


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.

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