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Supporting Hyperplane

A supporting hyperplane of a set K subset= R^n is a hyperplane through a boundary point of K such that K lies entirely in one of its closed half-spaces. More precisely, if x_0 is a boundary point and a is a nonzero vector with a·x<=a·x_0 for every x in K, then

H={x:a·x=a·x_0}

is a supporting hyperplane at x_0. The symbol · denotes the dot product (Boyd and Vandenberghe 2004, pp. 50-51).

In three dimensions, a supporting hyperplane is a supporting plane. Every boundary point of a nonempty set that is a convex set has a supporting hyperplane, but the supporting hyperplane need not be unique (Boyd and Vandenberghe 2004, p. 51).

See also

Boundary Point, Convex Body Support Function, Convex Set, Half-Space, Hyperplane, Plane

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References

Boyd, S. and Vandenberghe, L. "Supporting Hyperplanes." §2.5.2 in Convex Optimization. Cambridge, England: Cambridge University Press, pp. 50-51, 2004. https://doi.org/10.1017/CBO9780511804441. https://web.stanford.edu/~boyd/cvxbook/.

Cite this as:

Weisstein, Eric W. "Supporting Hyperplane." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SupportingHyperplane.html

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