1

Let $K_n(r) = [-r,r]\times[-r,r]\times\cdots\times[-r,r] \in\mathbb{R}^n$ be the $n$-Box of edge length $2r$. I think we can uniquely partition $K_n(1)$ into $2^n$ disjoint n-dimensional cubes, each of which are translates of $K_n(1/2)$. If so, then draw the largest $n$-Balls inscribed inside each of them. Now consider the central $n$-Ball which touches all these other balls externally. Let the radius of this ball be $r_n$. Below is an example when n=2. The coloured circle is the central circle, with radius $r_2=\frac{\sqrt{2}-1}{2}$. It can be derived easily that for a general $n\in\mathbb{N}$ we have $r_n = \frac1 2(\sqrt{n}-1)$. (derivation)

$^{\text{for n=2, }r_n \cong 0.2071}$

Intuitively it seems that the central ball should always stay inside $K_n$. But for $n>9$, we get $$r_n = \frac12(\sqrt{n}-1) >\frac12(\sqrt9-1)=1$$

Doesn't this imply that the central ball is no more contained inside $K_n$ for $n>9$? Clearly I've made some mistake in my arguments. Can someone point it out? Thanks!

Potemkin Metro Card
  • 1,598
  • 2
    I assume the mistake is in the intuition rather than the argument. Notice that already in 3D you can see how there is some gap in between the $2^3$ inscribed balls through which the central ball can "see the outside". It's not hard to imagine that in higher dimensions this gap could become big enough for the central ball to poke out of. –  May 19 '17 at 16:59

0 Answers0