Abstract
We prove that there is a universal constant C > 0 with the following property. Suppose that n ∈ ℕ and that A = (aij) ∈ Mn(ℝ) is a symmetric stochastic matrix. Denote the second-largest eigenvalue of A by λ2(A). Then for any finite-dimensional normed space (X, ⊥ · ⊥) we have (Equation presented) It follows that if an n-vertex O(1)-expander embeds with average distortion D ≧ 1 into X, then necessarily dim(X) ≳ nc/D for some universal constant c > 0. This is sharp up to the value of the constant c, and it improves over the previously best-known estimate dim(X) ≳ (log n)2/D2 of Linial, London and Rabinovich, strengthens a theorem of Matoušek, and answers a question of Andoni, Nikolov, Razenshteyn and Waingarten.
Original language | English (US) |
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Title of host publication | 33rd International Symposium on Computational Geometry, SoCG 2017 |
Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |
Pages | 501-5016 |
Number of pages | 4516 |
Volume | 77 |
ISBN (Electronic) | 9783959770385 |
DOIs | |
State | Published - Jun 1 2017 |
Event | 33rd International Symposium on Computational Geometry, SoCG 2017 - Brisbane, Australia Duration: Jul 4 2017 → Jul 7 2017 |
Other
Other | 33rd International Symposium on Computational Geometry, SoCG 2017 |
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Country/Territory | Australia |
City | Brisbane |
Period | 7/4/17 → 7/7/17 |
All Science Journal Classification (ASJC) codes
- Software