Large convex cones in hypercubes

A family of subsets of [n] is positive linear combination free if the characteristic vector of neither member is the positive linear combination of the characteristic vectors of some other ones. We construct a positive linear combination free family which contains (1-o(1))2ⁿ subsets of [n] and we give tight bounds on the o(1)2ⁿ term. The problem was posed by Ahlswede and Khachatrian [Cone dependence—a basic combinatorial concept, Preprint 00-117, Diskrete Strukturen in der Mathematik SFB 343, Universität Bielefeld, 2000] and the result has geometric consequences.     

Remarks on convex cones

We point out in this note that the class of cones in a locally convex topological vector space satisfying property () or piecewise relatively weakly compact cones is exactly the class of cones admitting weakly compact bases or the class of cones whose closures admit weakly compact bases.This work was supported by a Monash University Postdoctoral Fellowship.     

Nondegenerate families of convex cones and convex polytopes

This paper describes relations between convex polytopes and certain families of convex cones in R ⁿ .The purpose is to use known properties of convex cones in order to solve Helly type problems for convex sets in R ⁿ or for spherically convex sets in S _n , the n-dimensional unit sphere. These results are strongly related to Gale diagrams.     

Weak compactness in locally convex cones

Positivity - Using the coarsest weak topologies, we present the necessary and sufficient conditions for the weak upper, lower and symmetric compactness of subsets in cones. This leads us to...     

Solidity indices for convex cones

The issue addressed in this work is how to measure the degree of solidity of a closed convex cone in the Euclidean space ${\mathbb{R}^n}$ . One compares and establishes all sort of relations between the metric, the volumetric, and the Frobenius solidity indices.     

Duality on locally convex cones

The theory of locally convex cones as a branch of functional analysis was presented by K. Keimel and W. Roth in [K. Keimel, W. Roth, Ordered Cones and Approximation, Lecture Notes in Math., vol. 1517, Springer-Verlag, Heidelberg, 1992]. We study some more results about dual cones and adjoint operators on locally convex cones. Moreover we introduce the concept of the uniformly precompact sets and discuss their relations with σ-bounded sets. Some results obtained about inductive limit, projective limit, metrizability and quotients of locally convex cones.     

Locally convex quotient lattice cones

Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones.     

Completeness on locally convex cones

We investigate complete and compact subsets for the lower, upper and symmetric topologies of a locally convex cone and prove that weakly closed sets will be weakly compact, whenever they are weakly precompact. This leads to the weak* compactness of the polars of neighborhoods and weak compactness of the lower, upper and symmetric neighborhoods.     

Radial symmetry and partially overdetermined problems in a convex cone

We obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function P and integral identities. In dimension 2, we prove Serrin-type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone.     

Bifractional inequalities and convex cones

In this paper we give a systematic study of a class of linear inequalities related to convex cones in linear spaces. In particular, Chebyshev and Andersson type inequalities are discussed. Some classical and new inequalities are derived from the results.     

Minimax tests for convex cones

Let (P : R ^p ) be a simple shift family of distributions onR ^p , and letK R ^p be a convex cone. Within the class of nonrandomized tests ofK versusR ^p K, whose acceptance regionA satisfiesA=A+K, a test with minimal bias is constructed. This minimax test is compared to a likelihood ratio type test, which is optimal with respect to a different criterion. The minimax test is mimicked in the context of linear regression and one-sided tests for covariance matrices.