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Geometric Inequality - ILL 1990 ITA2

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September 18, 2010
inequalitiesgeometry unsolvedgeometry

Problem Statement

Let VV be a finite set of points in three-dimensional space. Let S1,S2,S3S_1, S_2, S_3 be the sets consisting of the orthogonal projections of the points of VV onto the yzyz-plane, zxzx-plane, xyxy-plane, respectively. Prove that ∣V∣2≤∣S1∣⋅∣S2∣⋅∣S3∣| V|^2 \leq | S1|\cdot|S2|\cdot |S3|, where ∣A∣| A| denotes the number of elements in the finite set A.A.
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