Problem 2

Consider a convex polyhedron

P_1

with nine vertices

A_1,A_2,\ldots,A_9

. Let

P_i

be obtained from

P_1

by a translation that moves

A_1

A_i

(

i=2,3,\ldots,9

). Prove that at least two of

P_1,P_2,\ldots,P_9

have an interior point in common.

Step 4 of 4: Compare volumes and force overlap

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\operatorname{vol}(2P_1)=2^3V=8V<9V=\sum_{i=1}^9\operatorname{vol}(P_i)

Detailed analysis

Let $V=\operatorname{vol}(P_1)>0$ . A factor-two dilation in three dimensions has volume $8V$ , whereas the nine translates have total volume $9V$ . If their interiors were pairwise disjoint, their union would have volume $9V$ (boundary overlaps have volume zero), yet it is contained in $2P_1$ of volume $8V$ , a contradiction. Thus at least two have a common interior point.