The main purpose of this paper is to introduce the compatibility of $M$-fuzzifying topologies and $M$-fuzzifying convexities, define an $M$-fuzzifying topological convex space, and give a method to generate an $M$-fuzzifying topological convex space. Some characterizations of $M$-fuzzifying topological convex spaces are presented. Finally, the notion of $M$-fuzzifying weak topologies is obtaine...

This paper deals with inequalities for the volume of a convex body and the volume of the projection body, the L-centroid body, and their polars. Examples are the Blaschke-Santaló inequality, the Petty and Zhang projection inequalities, the Busemann-Petty inequality. Other inequalities of the same type are still at the stage of conjectures. The use of special continuous movements of convex bodie...

Affine isoperimetric inequalities compare functionals, associated with convex (or more general) bodies, whose ratios are invariant under GL(n)-transformations of the bodies. These isoperimetric inequalities are more powerful than their better-known relatives of a Euclidean flavor. To be a bit more specific, this article deals with inequalities for centroid and projection bodies. Centroid bodies...

in this paper, we define the notions of fuzzy congruence relations and fuzzy convex subalgebras on a commutative residuated lattice and we obtain some related results. in particular, we will show that there exists a one to one correspondence between the set of all fuzzy congruence relations and the set of all fuzzy convex subalgebras on a commutative residuated lattice. then we study fuzzy...

Hermite-Hadamard inequality is one of the fundamental applications of convex functions in Theory of Inequality. In this paper, Hermite-Hadamard inequalities for $mathbb{B}$-convex and $mathbb{B}^{-1}$-convex functions are proven.

Let K be a convex body in R, i.e. a compact convex set with a non-empty interior. We say that K is origin-symmetric if K = −K. The presence of origin-symmetry is an essential assumption in various problems. Many results that hold for origin-symmetric convex bodies fail in the absence of the symmetry condition. For example, origin-symmetric convex bodies are uniquely determined by the volumes of...

We extend a theorem of Groemer’s on the expected volume of a random polytope in a convex body. The extension involves various ways of generating random convex sets. We also treat the case of absolutely continuous probability measures rather than convex bodies. As an application, we obtain a new proof of a recent result of Lutwak, Yang and Zhang on the volume of Orlicz-centroid bodies.

which is still the best known estimate for arbitrary convex bodies. Another class consists of convex bodies with small volume ratio with respect to the ellipsoid of minimal volume. This includes the class of zonoids. K. Ball [BA] solved the problem for the duals of zonoids, i.e. unit balls of subspaces of an L1-space, briefly L1-sections. Theorem 1 (K. Ball) For a convex, symmetric body K ⊂ IR ...

For a convex body K, let us denote by t(K) the largest number for which there exists a packing with finitely many translates of K in which every translate has at least t(K) neighbours. In this paper we determine t(K) for convex discs and 3-dimensional convex cylinders. We also examine how small the cardinalities of the extremal configurations can be in these cases. MSC 2000: 52C17, 52A10

Topological means are used for the study of approximation of 2-dimensional sections of a 3-dimensional convex body by affine-regular pentagons and approximation of a centrally symmetric convex body by a prism. Also, the problem of estimating the relative surface area of the sphere in a normed 3-space, the problem on universal covers for sets of unit diameter in Euclidean space, and some related...