I'm interested in learning how the geometry of the euclidean plane, with its problems concerning points, lines, circles, conics or higher curves, can or cannot be extended to infinite dimensions.

I understand that there are infinite-dimensional sequence-spaces and function-spaces, but the books of introductory functional analysis i know of seem to deal with different kinds of problems.

To be more specific, here are some examples concerning what i'm interested in:

In 1,2,3,4,... and all finite dimensions, the point (1,1,...,1) is well-defined. In infinite dimensions, its length is infinite, so does the point exist at all?

Also, finite dimensions also mean discrete dimensions, so we can visualize coordinate axes, up to 3, and even imagine that axes do exist for higher countable dimensions. But in spaces of real functions where dimensions are uncountable, does the concept of axis and direction still apply?

Hoping to make clearer what i'm asking for:
Triangles, circles, conics, cubes, spheres, are they useful concepts in a function space?
Also, constructions with ruler and compass?
Maybe of "platonic solids" in infinite dimensions?

Are there books that present a coherent approach along these lines?

  • 4
    $\begingroup$ The keyword is "Geometry of Banach spaces" $\endgroup$ – Giuseppe Negro Sep 27 '18 at 8:38
  • $\begingroup$ "In 1,2,3,4,... and all finite dimensions, the point (1,1,...,1) is well-defined." -- I disagree. This only makes sense if you have a canonical basis. There are many vector spaces in finite dimensions without a canonical or obvious choice of basis. (the simplest I can think of right now is the de Rahm cohomology space $H^k(M)$ for some manifold $M$ and $k>0$) $\endgroup$ – s.harp Sep 27 '18 at 15:35

Regular solids in finite spaces with more than 4 dimensions will be simplex, crosspolytope (aka orthoplex), and hypercube.

For the unit edged regular simplex within dimension $D$ we have:
$$circumradius = \sqrt{\frac{D}{2(D+1)}}$$ $$inradius = \frac{1}{\sqrt{2D(D+1)}}$$ $$height = \sqrt{\frac{D+1}{2(D+d)}} \ \ \ \ in \ orient. acc. to \ \ d||(D-1-d)$$ $$volume = \frac{1}{D!}\sqrt{\frac{D+1}{2D}}$$ $$dihedral \ angle = \arccos(\frac{1}{D})$$ thus, an infinite dimensional regular simplex becomes right angled and will have a zero sized inradius!

For the unit edged hypercube within dimension $D$ we have: $$circumradius = \frac{\sqrt{D}}{2}$$ $$inradius = \frac{1}{2}$$ $$volume = 1$$ $$dihedral \ angle = 90°$$ but, as you already said, the transition to infinity here would be diverging / undefined.

For the unit edged orthoplex within dimension $D$ we have: $$circumradius = \frac{1}{\sqrt{2}}$$ $$inradius = \frac{1}{\sqrt{2D}}$$ $$volume = \frac{\sqrt{2D}}{D!}$$ $$dihedral \ angle = \arccos(\frac{2}{D} - 1)$$ thus here the transition towards infinity provides both the inradius and the volume becoming zero, while the dihedral angle becomes $180°$. That is, this figure there would become a flat honeycomb! And, as its circumradius still is finite, we even get a bounded honeycomb there!

--- rk


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