论文标题
在一般倒数$σ_K$方程的凸度上
On the Convexity of General Inverse $σ_k$ Equations
论文作者
论文摘要
我们证明，如果一组水平$ n $ enverse unverse $σ_k$方程$ f（λ_1，\ cdots，λ_n）=λ_1\cdotsλ_n-\ sum_ \ sum_ {k = 0}^{n -1}^{n -1} \ mathbb {r}^n $，其中$ c_k $是实际数字不需要的，而$γ_n$是正矫正的，那么此级别集合为凸。 As an application, this result justifies the convexity of the level set of all general inverse $σ_k$ type equations, for example, the Monge--Ampère equation, the Hessian equation, the J-equation, the deformed Hermitian--Yang--Mills equation, the special Lagrangian equation, etc. Moreover, we find a numerical condition to verify whether a level set of a general inverse $σ_k$ equation is包含在$ q +γ_n$中的某些$ q \ in \ mathbb {r}^n $中，这是确定此级别集合的凸度的一种方法。
We prove that if a level set of a degree $n$ general inverse $σ_k$ equation $f(λ_1, \cdots, λ_n) = λ_1 \cdots λ_n - \sum_{k = 0}^{n-1} c_k σ_k(λ) = 0$ is contained in $q + Γ_n$ for some $q \in \mathbb{R}^n$, where $c_k$ are real numbers not necessary to be non-negative and $Γ_n$ is the positive orthant, then this level set is convex. As an application, this result justifies the convexity of the level set of all general inverse $σ_k$ type equations, for example, the Monge--Ampère equation, the Hessian equation, the J-equation, the deformed Hermitian--Yang--Mills equation, the special Lagrangian equation, etc. Moreover, we find a numerical condition to verify whether a level set of a general inverse $σ_k$ equation is contained in $q + Γ_n$ for some $q \in \mathbb{R}^n$, which is a way to determine the convexity of this level set.
