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Interpolation of random hyperplanes. (English) Zbl 1127.60008

Summary: Let \(\{(Z_i,W_i):i=1,\dots,n\}\) be uniformly distributed in \([0,1]^d\times \mathbb{G}(k,d)\), where \(\mathbb{G}(k,d)\) denotes the space of \(k\)-dimensional linear subspaces of \(\mathbb{R}^d\). For a differential function \(f:[0,1]^k\to[0,1]^d\), we say that \(f\) interpolates \((z,w)\in[0,1]^d \times\mathbb{G}(k,d)\) if there exists \(x\in[0,1]^k\) such that \(f(x)=z\) and \(\vec f(x)=w\), where \(\vec f(x)\) denotes the tangent space at \(x\) defined by \(f\). For a smooth class \({\mathcal F}\) of Hölder type, we obtain probability bounds on the maximum number of points a function \(f\in{\mathcal F}\) interpolates.

MSC:

60D05 Geometric probability and stochastic geometry
62G10 Nonparametric hypothesis testing
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