24/03/2010, 01:41
If \( \displaystyle x=(x_1,x_2,x_3)\in \mathbb{R}^3 \) and \( \displaystyle \Pi \subseteq \mathbb{R}^3 \) is the plane of equation \( \displaystyle ay_1+by_2+cy_3=\alpha \) , then the point-plane distance from \( \displaystyle x \) to \( \displaystyle \Pi \) is:
\( \displaystyle \text{dist} (x,\Pi )=\frac{|ax_1+bx_2+cx_3-\alpha|}{\sqrt{a^2+b^2+c^2}} \) .
18/04/2010, 02:52
gugo82 ha scritto:Exercise:
Let \( \displaystyle (X,||\cdot ||) \) an infinite-dimensional normed vector space and \( \displaystyle \Pi=\Pi_{u,\alpha} \subseteq X \) an affine hyperplane.
1. Prove that for each \( \displaystyle x\in X \) we have:
(*) \( \displaystyle \text{dist} (x,\Pi) = \frac{|\langle u,x\rangle -\alpha|}{||u||_*} \) .Testo nascosto, fai click qui per vederloHints: 1. Use the definition of \( \displaystyle ||u||_* \) and homogeneity to determine which values of \( \displaystyle r>0 \) make \( \displaystyle r\mathcal{B} \cap \Pi_{u,||u||_*} =\emptyset \) and which don't; use the translation invariance of the distance function to complete the proof.
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