On the Sharpness of One Integral Inequality for Closed Curves in ℝ^{4}

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
$\int_\gamma\sqrt{k_1^2+k_2^2+k_3^2}\,ds\geq 2\sqrt{5}\pi$.