Euclidean Quotients of Finite Metric Spaces

نویسندگان

  • Manor Mendel
  • Misha Gromov
چکیده

This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M and α ≥ 1, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embedings into lp, and the particular case of the hypercube. “ Our approach to general metric spaces bears the undeniable imprint of early exposure to Euclidean geometry. We just love spaces sharing a common feature with Rn .” Misha Gromov.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Assessment of the Log-Euclidean Metric Performance in Diffusion Tensor Image Segmentation

Introduction: Appropriate definition of the distance measure between diffusion tensors has a deep impact on Diffusion Tensor Image (DTI) segmentation results. The geodesic metric is the best distance measure since it yields high-quality segmentation results. However, the important problem with the geodesic metric is a high computational cost of the algorithms based on it. The main goal of this ...

متن کامل

Extended graphs based on KM-fuzzy metric spaces

This paper,  applies the concept  of KM-fuzzy metric spaces and  introduces a novel concept of KM-fuzzy metric  graphs based on KM-fuzzy metric spaces.  This study, investigates the finite KM-fuzzy metric spaces with respect to metrics and KM-fuzzy metrics and constructs KM-fuzzy metric spaces on any given non-empty sets. It tries to  extend   the concept of KM-fuzzy metric spaces to  a larger ...

متن کامل

On the metric triangle inequality

A non-contradictible axiomatic theory is constructed under the local reversibility of the metric triangle inequality. The obtained notion includes the metric spaces as particular cases and the generated metric topology is T$_{1}$-separated and generally, non-Hausdorff.

متن کامل

Currents in Metric Spaces

We develop a theory of currents in metric spaces which extends the classical theory of Federer–Fleming in euclidean spaces and in Riemannian manifolds. The main idea, suggested in [20, 21], is to replace the duality with differential forms with the duality with (k+ 1)-ples (f, π1, . . . , πk) of Lipschitz functions, where k is the dimension of the current. We show, by a metric proof which is ne...

متن کامل

Bilipschitz Embeddings of Metric Spaces into Euclidean Spaces

When does a metric space admit a bilipschitz embedding into some finite-dimensional Euclidean space? There does not seem to be a simple answer to this question. Results of Assouad [A1], [A2], [A3] do provide a simple answer if one permits some small (“snowflake”) deformations of the metric, but unfortunately these deformations immediately disrupt some basic aspects of geometry and analysis, lik...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2003