We investigate the uniqueness for the Monge–Ampère type equation

where u is the restriction of the support function on the sphere \({\mathbb {S}}^{n-1}\), of a convex body that contains the origin in its interior and \(G:(0,\infty )\rightarrow (0,\infty )\) is a continuous function. The problem was initiated by Firey (Mathematika 21(1): 1–11, 1974) who, in the case \(G(\theta )=\theta ^{-1}\), asked if \(u\equiv 1\) is the unique solution to (1). Recently, Brendle et al. (Acta Mathe 219(1): 1–16, 2017) proved that if \(G(\theta )=\theta ^{-p}\), \(p>-n-1\), then u has to be constant, providing in particular a complete solution to Firey’s problem. Our primary goal is to obtain uniqueness (or nearly uniqueness) results for (1) for a broader family of functions G. Our approach is very different than the techniques developed in Brendle et al. (2017).

我们研究了 Monge-Ampère 类型方程的唯一性

其中u是球体\({\mathbb {S}}^{n-1}\)上的支持函数的限制，该球体的内部包含原点和\(G:(0,\ infty )\rightarrow (0,\infty )\)是一个连续函数。这个问题是由 Firey (Mathematika 21(1): 1–11, 1974) 发起的，他在这种情况下\(G(\theta )=\theta ^{-1}\)询问是否\(u\equiv 1 \)是 (1) 的唯一解。最近，布伦德尔等人。(Acta Math 219(1): 1–16, 2017) 证明如果\(G(\theta )=\theta ^{-p}\) , \(p>-n-1\)，那么u必须是恒定的，特别是为 Firey 的问题提供完整的解决方案。我们的主要目标是为更广泛的函数G族获得 (1) 的唯一性（或接近唯一性）结果。我们的方法与 Brendle 等人开发的技术非常不同。(2017)。