2024/08/07 by Ayodeji Lindblad, Lindblad, Ayodeji · 1 citation
Engineering · Mathematics · #Manufacturing Process and Optimization #Mathematical Approximation and Integration #Optimization and Packing Problems
paper · pdf · doi:10.48550/arxiv.2408.04044
A spherical t-design curve was defined by Ehler and Gröchenig to be a continuous, piecewise smooth, closed curve on the sphere with finitely many self-intersections whose associated line integral applied to any polynomial of degree at most t evaluates to the average of this polynomial on the sphere. These authors posed the problem of proving that there exist sequences (γt)t=0^∞ of t-design curves on Sd of asymptotically optimal length ℓ(γt)\asymp td-1 as t→∞ and solved this problem for d=2. This work solves the problem for d=3 by proving that there exists a constant \mathscr C>0 such that for any C≥\mathscr C and t∈\Bbb N+, there exists a simple t-design curve on S3 of length Ct2.