In mathematics , the hypograph of a real-valued function denotes the set of all points that are on or below its graph .
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
definition
Be . The function's hypograph is defined by
![X \ subset \ R ^ n](https://wikimedia.org/api/rest_v1/media/math/render/svg/23a6b446fb9736703b3fe09ff010de5ef2e75f38)
![f \ colon X \ to \ R](https://wikimedia.org/api/rest_v1/media/math/render/svg/359ea801448b482438cb2149cfce6559dc3385b9)
![{\ displaystyle \ mathrm {hypo} \, f: = \ left \ {(x, \ mu) \ in X \ times \ mathbb {R} \,: \, \ mu \ leq f (x) \ right \} \ subseteq X \ times \ mathbb {R} \ ,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f897c352bc8f6a9abc827560b30c6462d39dbc63)
If the image space of the function is provided with a generalized inequality , then the hypograph is defined as
![\ preccurlyeq _ {K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bef9e870b49c8958943bdcc35e59a25b31466b7)
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.
properties
Be . The following applies to functions :
![X \ subset \ R ^ n](https://wikimedia.org/api/rest_v1/media/math/render/svg/23a6b446fb9736703b3fe09ff010de5ef2e75f38)
![f \ colon X \ rightarrow {\ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1265fb50a12cf56ecaf89f9046f65033690ccb3d)
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is concave if and only if the hypograph of forms a convex set.![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
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is above- half continuous if and only if the hypograph of forms a closed set.![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
- If an affine-linear function, then its hypograph defines a half-space in .
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
See also
Individual evidence
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^ Wilhelm Rödder, Peter Zörnig: Business Mathematics for Study and Practice 3 - Analysis II . Springer, 1997, ISBN 978-3-540-61716-7 , pp. 55 .