We find sufficient conditions for a probability measure μ to satisfy an inequality of the type ∫ Rd fF ( f ∫ Rd f 2 dμ ) dμ ≤ C ∫ Rd f2c∗ ( |∇f | |f | ) dμ + B ∫ Rd f dμ, where F is concave and c (a cost function) is convex. We show that under broad assumptions on c and F the above inequality holds if for some δ > 0 and ε > 0 one has ∫ ε 0 Φ ( δc [ tF ( 1t ) Iμ(t) ]) dt <∞, where Iμ is the isop...

In the field of optimal transport theory, an optimal map is known to be a gradient map of a potential function satisfying cost-convexity. In this paper, the Jacobian determinant of a gradient map is shown to be log-concave with respect to a convex combination of the potential functions when the underlying manifold is the sphere and the cost function is the distance squared. The proof uses the n...

The dynamic maintenance of the convex hull of a set of points in the plane is one of the most important problems in computational geometry. We present a data structure supporting point insertions in amortized O(log n · log log logn) time, point deletions in amortized O(log n · log logn) time, and various queries about the convex hull in optimal O(log n) worst-case time. The data structure requi...

The dynamic maintenance of the convex hull of a set of points in the plane is one of the most important problems in computational geometry. We present a data structure supporting point insertions in amortized O(log n · log log logn) time, point deletions in amortized O(log n · log logn) time, and various queries about the convex hull in optimal O(log n) worst-case time. The data structure requi...

in this paper, we propose and study exp-kumaraswamy distribution. some of its properties  are derived, including the density function, hazard rate function, quantile function, moments,  skewness  and kurtosis.   adata set isused to illustrate an application of the proposed distribution. also, we obtain a new distribution by transformation onexp-kumaraswamy distribution.   new distribution is an...

It is well-known that measures whose density the form e ? V where a uniformly convex potential on R n attain strong concentration properties. In search of notion log-concavity discrete hypercube, we consider { 1 , } multi-linear extension f satisfies log ? ? 2 ( x ) ? ? I for ? 0 which refer to as -semi-log-concave. We prove these satisfy nontrivial bound, namely, any Hamming Lipchitz test func...