We solve Talagrand’s entropy problem: the L2-covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension. This extends Dudley’s theorem on classes of {0, 1}valued functions, for which the shattering dimension is the VapnikChervonenkis dimension. In convex geometry, the solution means that the entropy of a convex body K is controlled by the maximal ...

BACKGROUND Pericardial fat has been implicated in the pathogenesis of obesity-related cardiovascular disease. Whether the associations of pericardial fat and measures of cardiac structure and function are independent of the systemic effects of obesity and visceral adiposity has not been fully explored. METHODS AND RESULTS Participants from the Framingham Heart Study (n=997; 54.4% women) under...

The landmark-based morphometric and meristic analysis of the kelp grouper (Epinephelus bruneus), red spotted grouper (E. akaara) and seven-banded grouper (E. septemfasciatus) were performed to compare the differentiation of overall body shape and structure. The measurements of the morphometric dimensions were observed in 25 parts (truss dimension: 16 parts; head part dimension: 9 parts) of 38 m...

We investigate some properties of coordinate projections. Among other results, we show that if a body K ⊂ Rn has an “almost extremal” volume ratio, then it has a projection of proportional dimension which is close to the cube. We also establish a sharp estimate on the shattering dimension of a convex hull of a class of functions.

TheNontrivial Projection Problem asks whether every finite-dimensional normed space admits a well-bounded projection of non-trivial rank and corank or, equivalently, whether every centrally symmetric convex body (of arbitrary dimension) is approximately affinely equivalent to a direct product of two bodies of non-trivial dimension. We show that this is true “up to a logarithmic factor.”

The Shattering dimension of a class is a real-valued version of the Vapnik-Chervonenkis dimension. We will present a solution to Talagrand’s entropy problem, showing that the L2-covering numbers of every uniformly bounded class of functions are exponential in the shattering dimension of the class. Formally we prove that there are absolute constants K and c such that for every 0 < t ≤ 1 and any ...

Automotive body styling appraisement is very important for body styling design. Because there is not tangible standard at the structure ascertaining of artificial neural network(ANN) and improvement of stability is needed for ANN inside black box characteristic, a bus-styling appraisement method using extension theory-based artificial neural network is presented. The key techniques of quasi-thr...

A convex body K in R is said to be reduced if the minimum width of each convex body properly contained in K is strictly smaller than the minimum width of K. We study the question of Lassak on the existence of reduced polytopes of dimension larger than two. We show that a pyramid of dimension larger than two with equal numbers of facets and vertices is not reduced. This generalizes the main resu...

We extend the results of [LMT] to the non-symmetric and quasiconvex cases. Namely, we consider finite-dimensional space endowed with gauge of either closed convex body (not necessarily symmetric) or closed symmetric quasi-convex body. We show that if a generic subspace of some fixed proportional dimension of one such space is isomorphic to a generic quotient of some proportional dimension of an...