It is NP-hard to approximate the hyperspherical radius of triangulated surfaces and triangulated high-dimensional spheres to within any almost-polynomial factor.
Lipschitz null-homotopy of mappings $S^3 \rightarrow S^2$
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abstract
This work focuses on important step in quantitative topology: given homotopic mappings from $S^m$ to $S^n$ of Lipschitz constant $L$, build the (asymptotically) simplest homotopy between them (meaning having the least Lipschitz constant). The present paper resolves this problem for the first case where Hopf invariant plays a role: $m = 3$, $n = 2$, constructing a homotopy with Lipschitz constant $O(L)$
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math.DG 1years
2019 1verdicts
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A hardness of approximation result in metric geometry
It is NP-hard to approximate the hyperspherical radius of triangulated surfaces and triangulated high-dimensional spheres to within any almost-polynomial factor.