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Generalizing the Johnson-Lindenstrauss lemma to k-dimensional affine subspaces

Alon Dmitriyuk, Yehoram Gordon (2009)

Studia Mathematica

Let ε > 0 and 1 ≤ k ≤ n and let W l l = 1 p be affine subspaces of ℝⁿ, each of dimension at most k. Let m = O ( ε - 2 ( k + l o g p ) ) if ε < 1, and m = O(k + log p/log(1 + ε)) if ε ≥ 1. We prove that there is a linear map H : m such that for all 1 ≤ l ≤ p and x , y W l we have ||x-y||₂ ≤ ||H(x)-H(y)||₂ ≤ (1+ε)||x-y||₂, i.e. the distance distortion is at most 1 + ε. The estimate on m is tight in terms of k and p whenever ε < 1, and is tight on ε,k,p whenever ε ≥ 1. We extend these results to embeddings into general normed spaces Y.

Geodesies on typical convex surfaces

Peter Manfred Gruber (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Using Baire categories uniqueness of geodesic segments and existence of closed geodesics on typical convex surfaces are investigated.

Geometric probabilities for non convex lattices. II

Andrei Duma, Marius Stoka (2002)

Bollettino dell'Unione Matematica Italiana

We solve problems of Buffon type for a lattice with elementary tile a nonconvex polygon, using as test bodies a line sigment and a circle.

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