Minimal Committee Problem for Inconsistent Systems of Linear Inequalities on the Plane

TL;DR

This paper introduces a novel geometric representation of inconsistent linear inequalities in the plane using polarity, and presents an algorithm to construct minimal committee solutions for such systems.

Contribution

It proposes a new point-based representation of linear inequalities via polarity and develops an algorithm for minimal committee solutions in inconsistent systems.

Findings

Algorithm effectively constructs minimal committee solutions.

Representation simplifies analysis of inconsistent systems.

Solutions to key problems are obtained through the proposed method.

Abstract

A representation of an arbitrary system of strict linear inequalities in R^n as a system of points is proposed. The representation is obtained by using a so-called polarity. Based on this representation an algorithm for constructing a committee solution of an inconsistent plane system of linear inequalities is given. A solution of two problems on minimal committee of a plane system is proposed. The obtained solutions to these problems can be found by means of the proposed algorithm.