On a representation of convex surfaces in \(n\)-dimensional spaces. (Q1432497)

The main results of the paper are contained in the following two theorems. Theorem 1. Suppose that there exists \(\varepsilon>0\) such that the function \(\rho(u):{K}_\varepsilon(\mathbb S_n)\to\mathbb R^1\) is continuous together with its partial derivatives up to the second order on \({K}_\varepsilon(\mathbb S_n)\) (where \({K}_\varepsilon(\mathbb S_n)\) is the union of all the balls of radius \(\varepsilon\) with their centers lying on unit \((n-1)\)-dimensional sphere \(\mathbb S_n\) in \(\mathbb R^n\)), the surface \(\Gamma(\mathbf u)=\Gamma(\rho,\mathbf u)=(x_1(\mathbf u),\dots,x_n(\mathbf u)):\mathbb S_n\to\mathbb R^n\) is defined by \[ x_i(\mathbf u)=\rho(\mathbf u)+\frac{\partial \rho(\mathbf u)}{\partial u_i}-u_i\sum_{j=1}^{n}\frac{\partial \rho(\mathbf u)}{\partial u_j}\,u_j,\quad i=1,2,\dots,n,\tag{*} \] and the quantity \[ K(\mathbf u)=\rho(\mathbf u)-\sum_{j=1}^{n}\frac{\partial \rho(\mathbf u)}{\partial u_j}\,u_j+\nabla\rho(\mathbf u)-\biggl(\sum_{j=1}^{n}\frac{\partial}{\partial u_j}\,u_j\biggr)^2\rho(\mathbf u) \] is positive almost everywhere on \((n-1)\)-sphere \(\mathbb S_n\). Then, (i) the \((n-1)\)-surface \(\Gamma(\mathbf u)\) is convex, closed and two-sided; (ii) the supporting plane of \(\Gamma\) at \(\Gamma(\mathbf u)\) has the form \[ \sum_{i=1}^n u_ix_i(\mathbf u)=\rho(\mathbf u); \] (iii) if \(\rho=\rho(\mathbf u)\) is nonnegative on \((n-1)\)-sphere \(\mathbb S_n\), then \(\rho(\mathbf u)\) is the support function for \(\Gamma\), and \(K(\mathbf u)\) is the Gaussian curvature of \(\Gamma\) at \(\Gamma(\mathbf u)\), i.e. \(\rho(\Gamma,\mathbf u)=\rho(\mathbf u)\) and \(K(\Gamma,\mathbf u)=K(\mathbf u)\); (iv) if \[ \rho_1(\mathbf u)=\rho(\mathbf u)+\sum_{i=1}^n c_i u_i, \] then \(\Gamma(\rho_1)=\Gamma(\rho)+\mathbf C\), where \(\mathbf C=(c_1,c_2,\ldots,c_n)\); (v) \(\Gamma(\rho+r)\) is the external \(r\)-equidistant surface of \(\Gamma(\rho,\mathbf u)\), and if \[ \rho(\mathbf u)-r-\sum_{j=1}^{n}\frac{\partial \rho(\mathbf u)}{\partial u_j}\,u_j+\nabla\rho(\mathbf u)-\biggl(\sum_{j=1}^{n}\frac{\partial}{\partial u_j}\,u_j\biggr)^2\bigl(\rho(\mathbf u)-r\bigr)>0 \] on the sphere \(\mathbb S_n\), then \(\Gamma(\rho-r,\mathbf u)\) is the internal \(r\)-equidistant surface of \(\Gamma(\rho,\mathbf u)\); (vi) the point \(\roman{grad}(\Gamma,\mathbf u)| _{u=0}\) is a Steiner point of the surface \(\Gamma\). Theorem 2. A necessary and sufficient condition for a surface \(\Gamma(\mathbf u)\) to be smooth, convex and two-sided is that (i) it is representable in the form (*), where \(\rho(\mathbf u)\) is an arbitrary function together with its partial derivatives up to the second order on \(K_\varepsilon(\mathbb S_n)\) for sufficiently small \(\varepsilon>0\), and (ii) \(K(\mathbf u)>0\) almost everywhere on the sphere \(\mathbb S_n\).