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::Supporting hyperplane

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A convex set <math>S</math> (in pink), a supporting hyperplane of <math>S</math> (the dashed line), and the half-space delimited by the hyperplane which contains <math>S</math> (in light blue).

In geometry, a supporting hyperplane of a set <math>S</math> in Euclidean space <math>\mathbb R^n</math> is a hyperplane that has both of the following two properties:

  • <math>S</math> is entirely contained in one of the two closed half-spaces bounded by the hyperplane
  • <math>S</math> has at least one boundary-point on the hyperplane.

Here, a closed half-space is the half-space that includes the points within the hyperplane.


Supporting hyperplane sections
Intro  Supporting hyperplane theorem  See also  References  

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A convex set <math>S</math> (in pink), a supporting hyperplane of <math>S</math> (the dashed line), and the half-space delimited by the hyperplane which contains <math>S</math> (in light blue).

In geometry, a supporting hyperplane of a set <math>S</math> in Euclidean space <math>\mathbb R^n</math> is a hyperplane that has both of the following two properties:

  • <math>S</math> is entirely contained in one of the two closed half-spaces bounded by the hyperplane
  • <math>S</math> has at least one boundary-point on the hyperplane.

Here, a closed half-space is the half-space that includes the points within the hyperplane.


Supporting hyperplane sections
Intro  Supporting hyperplane theorem  See also  References  

PREVIOUS: IntroNEXT: Supporting hyperplane theorem
<<>>