It is shown that for every subdivision of the d-dimensional Euclidean space, d ≥ 2, into n convex cells, there is a straight line that stabs at least Ω((log n/ log log n)1/(d−1)) cells. In other words, if a convex subdivision of d-space has the property that any line stabs at most k cells, then the subdivision has at most exp(O(kd−1 log k)) cells. This bound is best possible apart from a consta...

We show that for any 1 ≤ t ≤ c̃n log−5/2 n, the set of unconditional convex bodies in R contains a t-separated subset of cardinality at least exp ( exp ( c t2 log(1 + t) n )) . This implies the existence of an unconditional convex body in R which cannot be approximated within the distance d by a projection of a polytope with N faces unless N ≥ exp(c(d)n). We also show that for t ≥ 2, the cardina...

We introduce a new method of proving lower bounds on the depth of algebraic d-degree decision (resp. computation) trees and apply it to prove a lower bound (log N) (resp. (log N= log log N)) for testing membership to an n-dimensional convex polyhedron having N faces of all dimensions, provided that N > (nd) (n) (resp. N > n (n)). This bound apparently does not follow from the methods developed ...

Given a set S of n points on d, we show, for fixed d, how to construct in Ž . Ž . O n log n time a data structure we call the balanced aspect ratio BAR tree. A Ž . BAR tree is a binary space partition tree on S that has O log n depth in which Ž . every region is convex and ‘‘fat’’ that is, has a bounded aspect ratio . While previous hierarchical data structures such as k-d trees, quadtrees, oct...

Abstract In the preset study, we introduce new class of convex fuzzy-interval-valued functions which is called log- h -convex (log- FIVFs) by means fuzzy order relation. We have also investigated some properties FIVFs. Using this class, present Jensen and Hermite–Hadamard inequalities (HH-inequalities). Moreover, useful examples are presented to verify HH-inequalities for Several known special ...

A simple proof is given for the convexity of log det(I +KX) in the positive definite matrix variable X ≻ 0 with a given positive semidefinite K 0. Convexity of functions of covariance matrices often plays an important role in the analysis of Gaussian channels. For example, suppose Y and Z are independent complex Gaussian nvectors with Y ∼ N(0,K) and Z ∼ N(0,X). Then, I(Y;Y + Z) = log det(I +KX)...

Given a semi-convex potential V on convex and bounded domain Ω, we consider the Jordan–Kinderlehrer–Otto scheme for Fokker–Planck equation with V, which defines, fixed time step τ>0, sequence of densities ρk∈P(Ω). Supposing that is α-convex, i.e. D2V≥αI, prove Lipschitz constant logρ+V satisfies following inequality: Lip(log(ρk+1)+V)(1+ατ)≤Lip(log(ρk)+V). This provides exponential decay if α>0,...

We prove that the (B) conjecture and Gardner–Zvavitch are true for all log-concave measures rotationally invariant, extending previous results known Gaussian measures. Actually, our result apply beyond case of measures, instance, to Cauchy as well. For proof, new sharp weighted Poincaré inequalities obtained even probability with respect a invariant measure.

We prove a canonical bundle formula for generically finite morphisms in the setting of generalized pairs (with ?-coefficients). This complements Filipazzi’s with connected fibres. It is then applied to obtain subadjunction log centers pairs. As another application, we show that image an anti-nef pair has structure numerically trivial pair. readily implies result Chen–Zhang. Along way Shokurov t...