In this paper, we first characterize the convex $L$-subgroup of an $L$-ordered group by means of fourkinds of cut sets of an $L$-subset. Then we consider the homomorphic preimages and the product of convex $L$-subgroups.After that, we introduce an $L$-convex structure constructed by convex $L$-subgroups.Furthermore, the notion of the degree to which an $L$-subset of an $L$-ord...

Let K be a compact convex body in R, let Kn be the convex hull of n points chosen uniformly and independently in K, and let fipKnq denote the number of i-dimensional faces of Kn. We show that for planar convex sets, Erf0pKnqs is increasing in n. In dimension d ě 3 we prove that if limnÑ8 Erfd ́1pKnqs Anc “ 1 for some constants A and c ą 0 then the function n ÞÑ Erfd ́1pKnqs is increasing for n la...

‎let $x$ be a real normed  space, then  $c(subseteq x)$  is  functionally  convex  (briefly, $f$-convex), if  $t(c)subseteq bbb r $ is  convex for all bounded linear transformations $tin b(x,r)$; and $k(subseteq x)$  is  functionally   closed (briefly, $f$-closed), if  $t(k)subseteq bbb r $ is  closed  for all bounded linear transformations $tin b(x,r)$. we improve the    krein-milman theorem  ...

Convex optimization has the advantages on solving the problems with hundreds of variables and thousands of constraints. However, traditional multi-body model based zero moment point (ZMP) formulation makes it impossible to introduce the convex optimization into humanoid gait generation because the constraint can not be transferred into convex. This paper derives a new ZMP formulation by combini...

We elaborate on the use of shadow systems to prove a particular case of the conjectured lower bound of the volume product P(K) = minz∈int(K) |K|||K|, where K ⊂ R is a convex body and K = {y ∈ R : (y − z) · (x − z) 6 1 for all x ∈ K} is the polar body of K with respect to the center of polarity z. In particular, we show that if K ⊂ R is the convex hull of two 2-dimensional convex bodies, then P(...

It is proved that if C is a convex body in R" then C has an affine image C (of nonzero volume) so that if P is any 1-codimensional orthogonal projection, \PC\>\C\{tt~l)/n. It is also shown that there is a pathological body, K , all of whose orthogonal projections have volume about \fh~ times as large as |Ä"| . 0. Introduction The problems discussed in this paper concern the areas of shadows (or...

We introduce a new class of (not necessarily convex) bodies and show, among other things, that these bodies provide yet another link between convex geometric analysis and information theory. Namely, they give geometric interpretations of the relative entropy of the cone measures of a convex body and its polar and related quantities. Such interpretations were first given by Paouris and Werner fo...

In this paper, we consider the problem of packing two or more equal area polygons with disjoint interiors into a convex body K in E such that each of them has at most a given number of sides. We show that for a convex quadrilateral K of area 1, there exist n internally disjoint triangles of equal area such that the sum of their areas is at least 4n 4n+1 . We also prove results for other types o...

Let K be a convex body and K◦ its polar body. Call φ(K) = 1 |K||K◦| R

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K ⊂ Rd, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering...