Convex Subsets in Space

Suppose that $C$ is a convex subset of $\mathbb{R}^3$ with positive volume. Suppose that $C_1, C_2,\dots C_n$ are $n$ translated (not rotated) copies of $C$ such that $C_i \cap C \neq 0$ , but $C_i$ and $C_j$ intersect at most on the boundary for $i\neq j$ . Prove that $n \leq 27$ and that 27 is the best bound.

#SetNotation #Finn #CanYouProveIt? #ProveIt #ConvexSubsets

Easy Math Editor

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`bold` or `__bold__`	bold
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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$