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We know that for higher and higher dimensions the volume of a hyperball inscribed in a unit volume cube approaches zero. The ball is defined by the Euclidean distance.

Can you think of a mathematical form of an alternative distance measure, which would define a non-vanishing volume in increasing dimensions?

It should be some distance metric such that $d((x_1,\ldots,x_n),(0,\ldots,0))=1$ defines a shape and of course for all resulting coordinates $-0.5<=x_i<=0.5$ need to be in the cube. I guess it should also be symmetric in the variables and touch: $d(0.5,0,0,\ldots,0)=1$. Basically something to replace the sphere.

Gere
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1 Answers1

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$$d_\text{custom}(\mathbf{x},\mathbf{y}) := f(n) \cdot d_\text{euclid}(\mathbf{x},\mathbf{y})$$ with dimensionality $$n := \text{dim} \ \mathbf{x} = \text{dim} \ \mathbf{y}$$ and some (strongly) increasing function $f(n)$ should do the trick, hm?

Basically you want $$\lim_{n\rightarrow\infty} f(n)^{n} \frac{\pi^{n/2}}{\Gamma(n/2 + 1)} > 0$$ so $f(n) = n$ is sufficient.

GDumphart
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  • It still should be a maximal object inscribed in a unit cube. So scaling wouldn't do here. It should be something that extends into the corners more than a sphere. – Gere Nov 30 '14 at 15:22