Is there a geometric interpretation for the geometric mean of multiple numbers?

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Given a list of $n$ nonnegative real numbers $a_1, a_2, dots, a_n$, the geometric mean of that list is defined to be
$$sqrt[n]a_1a_2cdots a_n.$$



In the case of $n=2$, there are a few standard "geometric" interpretations of the geometric mean (generally involving power of a point, to visualize how the length $sqrta_1a_2$ depends on the lengths $a_1$ and $a_2$).



For the case of $nge 3$, are there are also nice geometric ways of understanding what the geometric mean measures? If so, what are they?




geometry means geometric-construction dimension-theory




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    Given a list of $n$ nonnegative real numbers $a_1, a_2, dots, a_n$, the geometric mean of that list is defined to be
    $$sqrt[n]a_1a_2cdots a_n.$$



    In the case of $n=2$, there are a few standard "geometric" interpretations of the geometric mean (generally involving power of a point, to visualize how the length $sqrta_1a_2$ depends on the lengths $a_1$ and $a_2$).



    For the case of $nge 3$, are there are also nice geometric ways of understanding what the geometric mean measures? If so, what are they?




    geometry means geometric-construction dimension-theory




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Given a list of $n$ nonnegative real numbers $a_1, a_2, dots, a_n$, the geometric mean of that list is defined to be
      $$sqrt[n]a_1a_2cdots a_n.$$



      In the case of $n=2$, there are a few standard "geometric" interpretations of the geometric mean (generally involving power of a point, to visualize how the length $sqrta_1a_2$ depends on the lengths $a_1$ and $a_2$).



      For the case of $nge 3$, are there are also nice geometric ways of understanding what the geometric mean measures? If so, what are they?




      geometry means geometric-construction dimension-theory




      share|cite|improve this question











      Given a list of $n$ nonnegative real numbers $a_1, a_2, dots, a_n$, the geometric mean of that list is defined to be
      $$sqrt[n]a_1a_2cdots a_n.$$



      In the case of $n=2$, there are a few standard "geometric" interpretations of the geometric mean (generally involving power of a point, to visualize how the length $sqrta_1a_2$ depends on the lengths $a_1$ and $a_2$).



      For the case of $nge 3$, are there are also nice geometric ways of understanding what the geometric mean measures? If so, what are they?





      geometry means geometric-construction dimension-theory





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      asked Aug 2 at 16:36









      Anonymous Reindeer

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          It's the side of an $n-$cube whose $n-$dimensional volume is equal to that of an $n-$box with sides $a_1,,dots,a_n$






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            Visual for 2nd and 3rd dimensions.
            – Simply Beautiful Art
            Aug 2 at 16:50










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          It's the side of an $n-$cube whose $n-$dimensional volume is equal to that of an $n-$box with sides $a_1,,dots,a_n$






          share|cite|improve this answer

















          • 2




            Visual for 2nd and 3rd dimensions.
            – Simply Beautiful Art
            Aug 2 at 16:50














          up vote
          1
          down vote













          It's the side of an $n-$cube whose $n-$dimensional volume is equal to that of an $n-$box with sides $a_1,,dots,a_n$






          share|cite|improve this answer

















          • 2




            Visual for 2nd and 3rd dimensions.
            – Simply Beautiful Art
            Aug 2 at 16:50












          up vote
          1
          down vote










          up vote
          1
          down vote









          It's the side of an $n-$cube whose $n-$dimensional volume is equal to that of an $n-$box with sides $a_1,,dots,a_n$






          share|cite|improve this answer













          It's the side of an $n-$cube whose $n-$dimensional volume is equal to that of an $n-$box with sides $a_1,,dots,a_n$







          share|cite|improve this answer













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          answered Aug 2 at 16:39









          saulspatz

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          10.5k21324







          • 2




            Visual for 2nd and 3rd dimensions.
            – Simply Beautiful Art
            Aug 2 at 16:50












          • 2




            Visual for 2nd and 3rd dimensions.
            – Simply Beautiful Art
            Aug 2 at 16:50







          2




          2




          Visual for 2nd and 3rd dimensions.
          – Simply Beautiful Art
          Aug 2 at 16:50




          Visual for 2nd and 3rd dimensions.
          – Simply Beautiful Art
          Aug 2 at 16:50












           

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