Quantitative combinatorial geometry for concave functions

Quantitative Combinatorial Geometry for Continuous Parameters

Exponentially concave functions and a new information geometry

Abstract. A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. It is known that gradient maps of exponentially concave functions are solutions of a MongeKantorovich optimal transport problem and allow for a better gradient approximation than those of ordinary concave functions. The approximation error, called L-diver...

The Sugeno fuzzy integral of concave functions

The fuzzy integrals are a kind of fuzzy measures acting on fuzzy sets. They can be viewed as an average membershipvalue of fuzzy sets. The value of the fuzzy integral in a decision making environment where uncertainty is presenthas been well established. Most of the integral inequalities studied in the fuzzy integration context normally considerconditions such as monotonicity or comonotonicity....

Basis functions for concave polygons

Polynomials suffice as finite element basis functions for triangles, parallelograms, and some other elements of little practical importance. Rational basis functions extend the range of allowed elements to the much wider class of well-set algebraic elements, where well-set is a convexity type constraint. The extension field from R(x, y) to R(x, y, √ x2 + y2) removes this quadrilateral constrain...

Divergence for s-concave and log concave functions

We prove new entropy inequalities for log concave and s-concave functions that strengthen and generalize recently established reverse log Sobolev and Poincaré inequalities for such functions. This leads naturally to the concept of f -divergence and, in particular, relative entropy for s-concave and log concave functions. We establish their basic properties, among them the affine invariant valua...