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Abstract We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of these results, we establish several generalizations of the celebrated Bang plank covering theorem. We prove a tight polynomial analog of the Bang theorem for the Euclidean ball and an even stronger polynomial version for the complex projective space. Specifically, for the ball, we show that for every real nonzero $$d$$-variate polynomial $$P$$ of degree $$n$$, there exists a point in the unit $$d$$-dimensional ball at distance at least $1/n$ from the zero set of the polynomial $$P$$. Using the polynomial approach, we also prove the strengthening of the Fejes Tóth zone conjecture on covering a sphere by spherical segments, closed parts of the sphere between two parallel hyperplanes. In particular, we show that the sum of angular widths of spherical segments covering the whole sphere is at least $$\pi $$.
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Lifting Methods in Mass Partition Problems
Abstract Many results about mass partitions are proved by lifting $$\mathds {R}^d$$ to a higher-dimensional space and dividing the higher-dimensional space into pieces. We extend such methods to use lifting arguments to polyhedral surfaces. Among other results, we prove the existence of equipartitions of $d+1$ measures in $$\mathds {R}^d$$ by parallel hyperplanes and of $d+2$ measures in $$\mathds {R}^d$$ by concentric spheres. For measures whose supports are sufficiently well separated, we prove results where one can cut a fixed (possibly different) fraction of each measure either by parallel hyperplanes, concentric spheres, convex polyhedral surfaces of few facets, or convex polytopes with few vertices.
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- Award ID(s):
- 2054419
- PAR ID:
- 10550269
- Publisher / Repository:
- Oxford Academic
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2023
- Issue:
- 16
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 14103 to 14130
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnac224
