Related papers: C*-Convexity of Norm Unit Balls

We consider the triangular ratio metric and estimate the radius of convexity for balls in some special domains and prove the inclusion relations of metric balls defined by the triangular ratio metric, the quasihyperbolic metric and the…

Let $M$ be a unital JB$^*$-algebra whose closed unit ball is denoted by $\mathcal{B}_M$. Let $\partial_e(\mathcal{B}_M)$ denote the set of all extreme points of $\mathcal{B}_M$. We prove that an element $u\in \partial_e(\mathcal{B}_M)$ is a…

In this note we prove two ellipsoid characterization theorems. The first one is that if $K$ is a convex body in a normed space with unit ball $M$, and for any point $p \notin K$ and in any 2-dimensional plane $P$ intersecting $\inter K$ and…

A number of results related to statistical classification on convex sets are presented. In particular, the focus is on the case where some of the covariates in the data and observation being classified can be missing. The form of the…

We prove that the $L_1$-norms associated with a positive element $a$ of a unital C*-algebra are equivalent to the norm of C*-algebra if and only if $a$ is invertible.

A pure geometric description of the Kobayashi balls of C-convex domains is given in terms of the so-called minimal basis.

For the $p$-harmonic function with strictly convex level sets, we find a test function which comes from the combination of the norm of gradient of the $p$-harmonic function and the smallest principal curvature of the level sets of…

Let n be a positive integer and c an n-tuple of natural numbers. A convex set in Euclidean n-space given by a family of linear relations in the elements of c and depending on their natural order is defined. The extremal elements of this…

We consider the covering of a ball in certain normed spaces by its congruent subsets and show that if the finite number of sets is not greater than the dimensionality of the space, then the centre of the ball either belongs to the interior…

We characterize projections among positive norm-one elements in unital C$^*$-algebras in pure geometric terms determined by the norm of the underlying Banach space. Concretely, let $A$ be a C$^*$-algebra (or a JB$^*$-algebra) whose positive…

Order unit property of a positive element in a $C^{*}$-algebra is defined. It is proved that precisely projections satisfy this order theoretic property. This way, unital hereditary $C^{*}$-subalgebras of a $C^{*}$-algebra are…

Under study are some vector optimization problems over the space of Minkowski balls, i.e., symmetric convex compact subsets in Euclidean space. A typical problem requires to achieve the best result in the presence of conflicting goals;…

We investigate the regularity of the marginals onto hyperplanes for sets of finite perimeter. We prove, in particular, that if a set of finite perimeter has log-concave marginals onto a.e. hyperplane then the set is convex.

For A a separable unital C*-algebra and M a separable McDuff II_1-factor, we show that the space Hom_w(A,M) of weak approximate unitary equivalence classes of unital *-homomorphisms A \rightarrow M may be considered as a closed, bounded,…

The problem of minimizing a multilinear function of binary variables is a well-studied NP-hard problem. The set of solutions of the standard linearization of this problem is called the multilinear set. We study a cardinality constrained…

Normed spaces appear to have very little going for them: aside from the hackneyed linear structure, you get a norm whose only virtue, aside from separating points, is the Triangle Inequality. What could you possibly prove with that? As it…

A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes…

We study the Hausdorff distance between a set and its convex hull. Let $X$ be a Banach space, define the CHD-module of space $X$ as the supremum of this distance for all subset of the unit ball in $X$. In the case of finite dimensional…

We present a necessary and sufficient condition for the reachable set, i.e., the set of states reachable from a ball of initial states at some time, of an ordinary differential equation to be convex. In particular, convexity is guaranteed…

Let $\Delta_m$ be the standard $m$-dimensional simplex of non-negative $m+1$ tuples that sum to unity and let $S$ be a nonempty subset of $\Delta_m$. A real valued function $h$ defined on a convex subset of a real vector space is $S$-almost…