Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0505426

A $K\_0$-avoiding dimension group with an order-unit of index two

Friedrich Wehrung
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that there exists a dimension group $G$ whose positive cone is not isomorphic to the dimension monoid Dim$L$ of any lattice $L$. The dimension group $G$ has an order-unit, and can be taken of any cardinality greater than or equal to $\aleph\_2$. As to determining the positive cones of dimension groups in the range of the Dim functor, the $\aleph\_2$ bound is optimal. This solves negatively the problem, raised by the author in 1998, whether any conical refinement monoid is isomorphic to the dimension monoid of some lattice. Since $G$ has an order-unit of index two, this also solves negatively a problem raised in 1994 by K.R. Goodearl about representability, with respect to $K\_0$, of dimension groups with order-unit of index 2 by unit-regular rings.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Friedrich Wehrung. 2005-06-01. A $K\_0$-avoiding dimension group with an order-unit of index two. https://arxiv.org/abs/math/0505426

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗