On the Christoffel-Minkowski problem of Firey's \(p\)-sum
\textit{W.~Firey} [Math. Scand. 10, 17--24 (1962; Zbl 0188.27303)] showed that for \(p>1\) the sum of the \(p\)-th powers of the support functions of two convex bodies containing the origin is also the \(p\)-th power of a support function. This fact was used later by [\textit{E.~Lutwak}, J. Differ. Geom. 38, No. 1, 131--150 (1993; Zbl 0788.52007); Adv. Math. 118, 244--294 (1996; Zbl 0853.52005)] to develop a Brunn-Minkowski theory for each \(p>1\), generalizing the classical case \(p=1\). In particular, there is an \(L^p\) version of the classical Minkowski problem, which has been studied by \textit{E.~Lutwak} and \textit{V.~Oliker} [J. Differ. Geom. 41, 227--246 (1995; Zbl 0867.52003)], and more recently by several other authors. Here the authors consider the \(L^p\) Christoffel-Minkowski problem. In the smooth category this is equivalent to finding solutions of \[ S_k(\nabla^2 u+g_0u)=u^{p-1}f \quad\text{on}\quad {\mathbb S}^n \leqno(*) \] satisfying the condition \[ \qquad \nabla^2 u+g_0u >0 \quad\text{on}\quad {\mathbb S}^n \leqno(**) \] for a given smooth positive function \(f\) on \({\mathbb S}^n\). Here \(k\in\{1,\dots,n-1\}\), \(g_0\) is the standard metric on \({\mathbb S}^n\) and the left hand side of \((*)\) denotes the \(k\)-th elementary symmetric function of the eigenvalues of \(\nabla^2 u+g_0u\). The positivity assumption on \(\nabla^2 u+g_0u\) is an unnatural restriction as far as the differential equation \((*)\) is concerned; it is needed to guarantee that \(u\) is indeed the support function of a convex body in \({\mathbb R}^{n+1}\). The main result of the paper is that for \(1\leq k<n\) and \(p>k+1\), the problem \((*)\) has a unique smooth \(k\)-admissible solution for any smooth positive function \(f\). Furthermore, \((**)\) holds provided \(\tilde f:=f^{-1/(p+k+1)}\) satisfies \(\nabla^2\tilde f+g_0\tilde f>0\) on \({\mathbf S}^n\). A variant of the above is also proved for \(p=k+1\).
scientific article; zbMATH DE number 2113880