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  1. If I have $\mathbb R^n$ an $p\in[ 1,\infty )$ and the distance between vectors v and w is defined as $\sqrt[p]{(|v_1-w_1|^p)}$.

    What does $d_\infty(v, w)$= max {$|v_1-w_1|,|v_2-w_2|,\ldots,|v_n-w_n|$} mean? How would you vizualize it?

  2. If I have $S^2\subset \mathbb R^3$ and $S^2$={ $v\in$ $\mathbb R^3$,|v|=1} then we conclude that is metric space. An observation: Shouldn't be enough to write $v\in$ $\mathbb R^2$? I'm confused why it is written in my textbook $v\in$ $\mathbb R^3$. Is it just random reason for example: It's ok, that third dimension of a vector doesn't matter or is a reason behind it?

  3. I somehow understand that it is possible to have more metrics in metric space- you just have two defined ways how to measure distance between elements. If I understand correctly, then for example by line or by arc. But then our professor told us that you only take the shortest way of measuring distance and that would be of course a line. Is this true?

I would really appreciate an explanation.

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continuity
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1 Answers1

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See Geometric interpretation of $x^3+y^3+z^3=k^3$, including the links in the reply. Suppose you want to draw a "circle" using $d_p=\sqrt[p]{\sum_i|v_i|^p}$ metric. Obviously for $p=2$ you get your usual circle. As you increase $p$, the circle would transform into a square with the sides parallel to the axes. In the limit $p\rightarrow\infty$ you can take the largest term in the sum outside of the root $p$. What you are left is with $1$ plus terms with absolute value less than $1$, raised to power of $\infty$. Those terms all become $0$. That's why your metric $d_\infty$ is described by the maximum value.

For part 2, $S$ is a cube centered on $0$, with side $2$.

For part 3, the answer is dependent on the metric. Once again, see this video

Andrei
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