MathLabs

Problem 2

Consider a convex polyhedron P1P_1 with nine vertices A1,A2,…,A9A_1,A_2,\ldots,A_9. Let PiP_i be obtained from P1P_1 by a translation that moves A1A_1 to AiA_i (i=2,3,…,9i=2,3,\ldots,9). Prove that at least two of P1,P2,…,P9P_1,P_2,\ldots,P_9 have an interior point in common.
Step 2 of 4: Use convexity to control sums
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p,q∈P1  ⟹  p+q2∈P1  ⟹  p+q∈2P1p,q\in P_1\implies \frac{p+q}{2}\in P_1\implies p+q\in2P_1
Detailed analysis

For any p,q∈P1p,q\in P_1, convexity gives (p+q)/2∈P1(p+q)/2\in P_1. Therefore p+q∈2P1p+q\in2P_1, where 2P1={2x:x∈P1}2P_1=\{2x:x\in P_1\} is the dilation of P1P_1 by factor 22 about the origin.