求解凸集空间上的多项式不等式及应用
Solving polynomial inequalities over spaces of convex sets and applications
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中文总结 AI 辅助
针对凸集空间的递归包含不等式方程组建立符号消元理论，证明其最小解的性质，应用于层叠包络并得出其半代数且可有效计算的结论。
中文摘要 AI 辅助
我们针对有限维实向量空间中未知量为凸子集的递归包含不等式有限方程组，建立了符号消元理论。方程右端为由变量和参数通过凸线性组合、有限并集以及编码严格凸组合的正几何并生成的形式表达式。我们证明每个参数赋值都有唯一最小的凸集值解，并给出一种有限的高斯消元型过程，该过程在消去未知量的同时保留此解，并生成其坐标集的仅含参数的表达式。更一般地，设$\boldsymbol{\beta}$为包含$\boldsymbol{\theta}$且在有限并、非负伸缩、闵可夫斯基和、正几何并及凸包下封闭的子集族。若所有参数集都属于$\boldsymbol{\beta}$，则最小解的每个坐标集都属于$\boldsymbol{\beta}$；当这些操作有效时，所得描述也有效。特别地，若参数是半多面体的有限并——其中半多面体是有界凸半线性集，等价于多面体相对内部的凸有限并——则每个坐标集都是半多面体，且可接受无量词半线性描述。我们将该理论应用于层叠包络。对于$V=U\bigoplus\bigoplus_{i=1}^{k}W_i$，$\boldsymbol{\text{dim}}\boldsymbol{W}_i=1$，$\boldsymbol{\text{Lambda}}=\bigcup_{i=1}^{k}(U+W_i)$，我们证明每个有限$\boldsymbol{S}\boldsymbol{\text{subset}}\boldsymbol{V}$的层叠包络$\boldsymbol{G}_{\boldsymbol{\text{Lambda}}}^{(\boldsymbol{\text{infinity}})}(\boldsymbol{S})$是半代数的，且可通过实数上的无量词公式有效计算。
英文摘要
We develop a symbolic elimination theory for finite systems of recursive containment inequalities whose unknowns are convex subsets of a finite-dimensional real vector space. The right-hand sides are formal expressions generated from variables and parameters by convex linear combinations, finite union, and a positive geometric join encoding strict convex combinations. We prove that every parameter assignment has a unique smallest convex-set-valued solution and give a finite Gaussian-elimination-type procedure that eliminates the unknowns while preserving this solution and produces parameter-only expressions for its coordinate sets. More generally, let $\mathcal B$ be a family of subsets containing $\emptyset$ and closed under finite unions, nonnegative dilation, Minkowski sums, positive geometric joins, and convex hulls. If all parameter sets lie in $\mathcal B$, then every coordinate set of the smallest solution lies in $\mathcal B$; when these operations are effective, so is the resulting description. In particular, if the parameters are finite unions of hemihedra---where a hemihedron is a bounded convex semi-linear set, equivalently a convex finite union of relative interiors of polytopes---then each coordinate set is a hemihedron and admits a quantifier-free semi-linear description. We apply this theory to lamination hulls. For \[ V=U\oplus\bigoplus_{i=1}^{k}W_i,\qquad \dim W_i=1,\qquad Λ=\bigcup_{i=1}^{k}(U+W_i), \] we prove that the lamination hull $G_Λ^{(\infty)}(S)$ of every finite $S\subset V$ is semi-algebraic and effectively computable by a quantifier-free formula over the reals.
