Let $E$ be a non-empty compact set and $H$ is any hyperplane. Show that $E$ has a supporting hyperplane parallel to $H$.
I have no idea to proceed the proof. Can anyone give me some hints? Thanks
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Hoan Do
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Did you forget to say that $E$ is convex or not ? Is the space euclidean ? – Tony Piccolo Aug 24 '15 at 12:30
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$E$ is not necessary to be convex. Space is Euclidean. Thanks – Hoan Do Aug 24 '15 at 13:35
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First of all remember that, in finite dimensions, every linear functional is continuous.
Let $p \cdot x=\alpha$ be an equation of $H$.
Since $E$ is compact, the continuous linear functional $x \rightarrow p \cdot x$ has a minimum $\beta$ and a maximum $\gamma$ in $E$.
Thus $\beta \le p \cdot x \le \gamma$ for every $x \in E$ and there exist $x_1,x_2\in E$ with $p \cdot x_1=\beta$ and $p \cdot x_2=\gamma$.
If $\beta < \gamma$, there are two supporting hyperplanes parallel to $H$.
If $\beta = \gamma$, the solution is unique ($E$ is contained in it).
All this paraphrased from M. Florenzano,C. Le Van Finite $\,$Dimensional Convexity and Optimization (2001), Proposition 2.1.2, p. 22 . On p. 21 you find the definitions you need.
Tony Piccolo
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