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 3 of 4: Contain every translate in 2P_1

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A_i\in P_1,\ x\in P_1\implies A_i+x\in2P_1\implies P_i\subseteq2P_1

Detailed analysis

Each $A_i$ is a vertex of $P_1$ , hence $A_i\in P_1$ . For every $x\in P_1$ , the point $A_i+x$ is a sum of two points of $P_1$ , so it lies in $2P_1$ . Since $P_i=A_i+P_1$ , this proves $P_i\subseteq2P_1$ for all nine translates.