METRIC THICKENINGS, BORSUK–ULAM THEOREMS, AND ORBITOPES
METRIC THICKENINGS, BORSUK–ULAM THEOREMS, AND ORBITOPES
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DOI：
10.1112/mtk.12010
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发表时间：
2019-07
期刊：
影响因子：
0.8
通讯作者：
Henry Adams;Johnathan Bush;F. Frick
中科院分区：
文献类型：
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作者：
Henry Adams;Johnathan Bush;F. Frick
度量空间的增厚捕捉了空间的局部几何性质。在这里，我们展示了应用程序的下界的拓扑结构的加厚的圆，更一般的领域。我们解释互连与几何圈行动的欧几里德空间，结构的零三角多项式，和定理的Borsuk-Ulam型。利用欧氏空间上圆作用轨道的凸船体的组合结构和几何结构，给出了圆的度量增厚的同伦型的几何证明。球面加厚的同伦连通性界允许我们证明在小直径集合上奇映射$S^n \to \mathbb{R}^{n+2}$的函数值的加权平均为零。我们证明了奇映射$S^{2n-1} \to \mathbb{R}^{2kn+2n-1}$的Borsuk-Ulam定理的另一个推广。我们证明了这样的结果，从圆的奇映射到任何欧几里德空间的最佳数量界。这反过来又意味着任何斜齐次三角多项式在特定直径的圆的子集上都有零;这些结果是最优的。
Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the geometry of circle actions on Euclidean space, the structure of zeros of trigonometric polynomials, and theorems of Borsuk-Ulam type. We use the combinatorial and geometric structure of the convex hull of orbits of circle actions on Euclidean space to give geometric proofs of the homotopy type of metric thickenings of the circle. Homotopical connectivity bounds of thickenings of the sphere allow us to prove that a weighted average of function values of odd maps $S^n \to \mathbb{R}^{n+2}$ on a small diameter set is zero. We prove an additional generalization of the Borsuk-Ulam theorem for odd maps $S^{2n-1} \to \mathbb{R}^{2kn+2n-1}$. We prove such results for odd maps from the circle to any Euclidean space with optimal quantitative bounds. This in turn implies that any raked homogeneous trigonometric polynomial has a zero on a subset of the circle of a specific diameter; these results are optimal.