论纽兰数与极对偶性
On Nieuwland Numbers and Polar Duality
AI总结：
该研究探讨凸多面体的纽兰数，将多面体穿越问题归约为二次约束可行性问题，证明八面体的纽兰数为3√2/4，并将纽兰数计算归约为半代数优化问题。
AI中文摘要：
若一个凸三维多面体可穿过其自身的一个复制体，则称其具有鲁珀特性质。凸多面体P的纽兰数是使得νP可穿过P的最大正实数ν。我们将P穿过Q的问题归约为二次约束集上的可行性问题，利用该结论证明八面体的纽兰数为3√2/4，且凸多面体的纽兰数计算可归约为固定变量数下多项式多个半代数优化问题。
英文摘要：
A convex 3D-polytope is said to have Rupert's property if it can pass through a copy of itself. The Nieuwland number of a convex polytope $P$ is the largest $ν\in \mathbb{R}^+$ such that $νP$ can pass through $P$. We reduce showing $P$ passes through $Q$ to a feasibility problem over a quadratic constraint set. Using this, we prove that the Nieuwland number of the octahedron is $\frac{3\sqrt2}{4}$ and that the computation of the Nieuwland number of a convex polytope can be reduced to polynomially many semialgebraic optimization problems in a fixed number of variables.
