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 1 of 4: Normalize the translations

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A_1=0,\qquad P_i=A_i+P_1

Detailed analysis

Translate the coordinate system so that $A_1=0$ . The translation carrying $A_1$ to $A_i$ has vector $A_i$ , hence $P_i=A_i+P_1$ for $i=1,\ldots,9$ , with $A_1=0$ giving $P_1$ itself.