### New structure of norms on R^n and their relations with the curvature of the plane curves

Amir Veisi, Ali Delbaznasab

#### Abstract

Let $f_1, f_2, \ldots, f_n$ be fixed nonzero real-valued functions on $\mathbb{R}$, the real numbers. Let $\varphi_n(X_n)= \big(x_1^2f_1^2+x_2^2f_2^2+ \ldots + x_n^2f_n^2 \big)^{\frac{1}{2}}$, where $X_n=(x_1, x_2, \ldots, x_n) \in \mathbb{R}^n$. We show that $\varphi_n$ has properties similar to a norm on the normed linear space. Although $\varphi_n$ is not a norm on $\mathbb{R}^n$ in general, it induces a norm on $\mathbb{R}^n$. For the nonzero function $F : \mathbb{R}^2 \to \mathbb{R}$, a curvature formula for the implicit curve $G(x, y)=F^2(x, y)=0$ at any regular point is given. A similar result is presented when $F$ is a nonzero function from $\mathbb{R}^3$ to $\mathbb{R}$. In continued, we concentrate on $F(x, y)=\int_a^b \varphi_2(x, y)dt$. It is shown that the curvature of $F(x, y)=c$ when $c>0$ is a positive multiple of $c^2$. Particularly, we observe that $F(x, y)=\int_0^{\frac{\pi}{2}} \sqrt{x^2 \cos^2 t + y^2 \sin^2 t} dt$ is an elliptic integral of the second kind.

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#### References

R. Goldman. Curvature formulas for implicit curves and surfaces. Computer Aided Geometric Design, 22(7):632–658, 2005.

C. G. Lekkerkerker. Geometry of numbers North-Holland Publishing Company, Amsterdam, 1969.

W. Rudin. Principles of Mathematical Analysis, McGraw-Hill, International Book Company, Ltd, 1976.

C. L. Siegel. Lecturcs on the geometry of numbers Springer-Verlag Berlin Heidelberg, 1989.

S. Willard. General Topology, Addison-Wesley, 1970.

DOI: http://dx.doi.org/10.23755/rm.v39i0.552

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