短球面t-设计曲线
Short spherical $t$-design curves
AI总结：
研究球面t-设计曲线的最小弧长，证明了明确谱下界，对特定\(t\)和维度是尖锐的，为\(t>1\)给出最优性结果，还为偶数维球构造曲线并在\(S^2\)中得到最短2-设计曲线候选。
AI中文摘要：
我们研究球面t-设计曲线的最小弧长，即\(S^d\)上的闭可求长曲线，其归一化弧长测度能精确积分次数至多为t的每个多项式。我们证明了一个明确的谱下界，在所有球中对\(t = 1\)以及在每个奇数维球中对\(t = 2\)是尖锐的，这为\(t>1\)的球面t-设计曲线给出了首个精确最优性结果。对于偶数维球，我们构造了2-设计曲线，其长度在\(d\to\infty\)时渐近匹配下界，在\(S^2\)中，我们使用数值优化和变分法得到最短2-设计曲线的一个候选。
英文摘要：
We study the minimum arclength of spherical $t$-design curves, i.e., closed rectifiable curves on $S^d$ whose normalized arclength measure exactly integrates every polynomial of degree at most $t$. We prove an explicit spectral lower bound that is sharp for $t=1$ in all spheres and for $t=2$ in every odd-dimensional sphere, yielding the first exact optimality results for spherical $t$-design curves with $t>1$. For even-dimensional spheres, we construct $2$-design curves whose lengths asymptotically match the lower bound as $d\to\infty$, and in $S^2$, we use numerical optimization and the calculus of variations to derive a candidate for the shortest $2$-design curve.
