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Polyhedral Geometry

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Optimization for Decision Making

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 137))

Abstract

A hyperplane in R n is the set of feasible solutions of a single linear equation. Let H = { x : a 1 x 1 + ⋯ + a n x n = a 0}, where the coefficient vector (a 1, …, a n )≠0, be a hyperplane in R n.

Only when n = 2 (i.e., in R 2 only) every hyperplane is a straight line, and vice versa. In Fig. 4.1, we show the hyperplane (straight line in R 2) corresponding to the equation x 1 + x 2 = 1.

When n ≥ 3, hyperplanes are not straight lines. Figure 4.2 shows a portion of the hyperplane corresponding to the equation x 1 + x 2 + x 3 = 1 in R 3.

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Authors and Affiliations

  1. Dept. Industrial and Operations Engineering, University of Michigan, 1205 Beal Avenue, Ann Arbor, MI, 48109-2117, USA

    Katta G. Murty (Professor)

  2. Systems Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    Katta G. Murty (Professor)

Authors
  1. Katta G. Murty
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Correspondence to Katta G. Murty .

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Murty, K.G. (2010). Polyhedral Geometry. In: Optimization for Decision Making. International Series in Operations Research & Management Science, vol 137. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1291-6_4

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