A Simple proof of Johnson-Lindenstrauss extension theorem

Johnson and Lindenstrauss proved that any Lipschitz mapping from an $n$-point subset of a metric space into Hilbert space can be extended to the whole space, while increasing the Lipschitz constant by a factor of $O(\sqrt{\log n})$. We present a simplification of their argument that avoids dimension reduction and the Kirszbraun theorem.Comment: 3 pages. Incorporation of reviewers' suggestion

The “Renewed†Kibbutz

Over the years, the unique form of cooperative settlement known as a Kibbutz, has had to undergo various difficulties and changes, yet it has managed to maintain its principles and survive. However, during the last decade, many kibbutzim have made substantial changes which contradict traditional Kibbutz ideology. This article is based on the report of the Public Committee for the kibbutzim, established in 2002 to recommend a new legal definition befitting the development that took place in the kibbutzim in the last decades. The Committee was also asked to submit its opinion on the issue of allocating apartments to kibbutz members. The article describes the inevitable changes that have occurred and analyzes the reasons and considerations that have led to the creation of a new form of Kibbutz.Agribusiness,

Truly Online Paging with Locality of Reference

The competitive analysis fails to model locality of reference in the online paging problem. To deal with it, Borodin et. al. introduced the access graph model, which attempts to capture the locality of reference. However, the access graph model has a number of troubling aspects. The access graph has to be known in advance to the paging algorithm and the memory required to represent the access graph itself may be very large. In this paper we present truly online strongly competitive paging algorithms in the access graph model that do not have any prior information on the access sequence. We present both deterministic and randomized algorithms. The algorithms need only O(k log n) bits of memory, where k is the number of page slots available and n is the size of the virtual address space. I.e., asymptotically no more memory than needed to store the virtual address translation table. We also observe that our algorithms adapt themselves to temporal changes in the locality of reference. We model temporal changes in the locality of reference by extending the access graph model to the so called extended access graph model, in which many vertices of the graph can correspond to the same virtual page. We define a measure for the rate of change in the locality of reference in G denoted by Delta(G). We then show our algorithms remain strongly competitive as long as Delta(G) >= (1+ epsilon)k, and no truly online algorithm can be strongly competitive on a class of extended access graphs that includes all graphs G with Delta(G) >= k- o(k).Comment: 37 pages. Preliminary version appeared in FOCS '9

Metric Cotype

We introduce the notion of metric cotype, a property of metric spaces related to a property of normed spaces, called Rademacher cotype. Apart from settling a long standing open problem in metric geometry, this property is used to prove the following dichotomy: A family of metric spaces F is either almost universal (i.e., contains any finite metric space with any distortion > 1), or there exists α > 0, and arbitrarily large n-point metrics whose distortion when embedded in any member of F is at least Ω((log n)^α). The same property is also used to prove strong non-embeddability theorems of L_q into L_p, when q > max{2,p}. Finally we use metric cotype to obtain a new type of isoperimetric inequality on the discrete torus

Some applications of Ball's extension theorem

We present two applications of Ball's extension theorem. First we observe that Ball's extension theorem, together with the recent solution of Ball's Markov type 2 problem due to Naor, Peres, Schramm and Sheffield, imply a generalization, and an alternative proof of, the Johnson-Lindenstrauss extension theorem. Second, we prove that the distortion required to embed the integer lattice {0,1,...,m}^n, equipped with the ℓ_p^n metric, in any 2-uniformly convex Banach space is of order min {n^(1/2 1/p),m^(1-2/p)}