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In $\Bbb {R^n}$ ,how can we find the length of the diagonal of a hypercube of side 1 for the usual known distances : (L2-norm ) or the euclidean distance and (L infinity norm using the max) ?

Adonis Hammoutene
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1 Answers1

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The diagonal $D$ can be written as $D = e_{1}+e_{2}+\cdots + e_{n}$, where $\{e_{1},...,e_{n}\}$ is the canonical basis for $\mathbb{R}^{n}$. Thus, with the usual Euclidean norm: $$||D|| = \sqrt{n}$$ Can you evaluate the other ones?

IamWill
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  • I unfortunately cannot evaluate the others but if we take the 3-hypercube which is cube ,I know that the length of the diagonal of the cube will be found using two times the Pythagore theorem, then we can found that the lenght of the diagonal is a×sqrt(3) where a denotes the length of its side . what do you think ? and thanks so much for answering this fast ! – Adonis Hammoutene Sep 22 '20 at 16:49
  • You are correct. If $a=1$, then the diagonal for a $3$d cube is simply $\sqrt{3}$. My answer is the generalization of this to $n$-dimensional case. Now, you have to evaluate the max norm. Do you know the definition of this norm? Note that the diagonal $D$ as written in my answer is equivalent to $D=(1,1,...,1)\in \mathbb{R}^{n}$. – IamWill Sep 22 '20 at 16:55
  • yeah thank you so much, I know its définition (sorry I might not know the term in english because i'm studiyng it in French – Adonis Hammoutene Sep 22 '20 at 16:57