Some New Characteristic Properties of the A-Pedal Hypersurfaces in $E^{n+1}$

Abstract

The primary purpose of this paper is to present the definition of the a-pedal hypersurface with respect to a point in the interior of a closed, convex and smooth hypersurface $M$. The secondary purpose of this paper is to give some new characteristic properties of the a-pedal hypersurfaces related to the support function, Gauss curvature, mean curvature, the first and second fundamental forms and their coefficients of $M$ (Section 3). Using the classical methods of the hypersurfaces in differential geometry we have established that the support function $h_{a}$ of the a-pedal hypersurface $M_{a}$ is equal to $\frac{h^{a+1}}{P_{a}}$ where $P_{a}^{2}=h^{2}+a^{2}\overset{III}{\nabla }(h,h)$.