Dimensionality Questions for ℓ ₁-Embeddings

Abstract

Given a distance space (X, d) which is ℓ ₁-embeddable, a natural question is to determine the smallest dimension m of an ℓ ₁-space EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeS4eHW2aa0 % baaSqaaiaaigdadaWgaaadbaaabeaaaSqaaiaad2gaaaGccqGH9aqp % caGGOaGaeSyhHe6aaWbaaSqabeaacaWGTbaaaOGaaiilaiaadsgada % WgaaWcbaGaeS4eHW2aaSbaaWqaaiaaigdaaeqaaaWcbeaakiaacMca % aaa!422C! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\ell _{{1_{}}}^m = ({\mathbb{R}^m},{d_{{\ell _1}}})$$ in which (X, d) can be embedded. A next question is whether there exists a finite point criterion for ℓ ^m₁ -embeddability, analogue to Menger’s result for the Euclidean space; this is the question of finding the order of congruence of ℓ ^m₁ . We present in this chapter several results related to these questions. Unfortunately fairly little is known. For instance, the order of congruence of ℓ ^m₁ is known only for m ≤ 2 and it is not even known whether ℓ ^m₁ has a finite order of congruence for m ≥ 3. More precisely, a distance space (X, d) is ℓ ¹₁ - (resp. ℓ ²₁ -embeddable) if and only if the same holds for every subspace of (X, d) on 4 (resp. on 6) points. As a consequence, one can recognize in polynomial time whether a distance is ℓ ^m₁ -embeddable for m ≤ 2. On the other hand, the complexity of checking ℓ ^m₁ -embeddability is not known for m ≥ 3. These results are presented in Section 11.1. A crucial tool for the proofs is the notion of ‘totally decomposable distance’ studied by Bandelt and Dress [1992]; Section 11.1.2 contains the facts about total decomposability that are needed for our treatment. Then we consider in Section 11.2 some bounds for the minimum ℓ _p -dimension of an arbitrary ℓ _p -embeddable distance space on n points.