MathLabs

Problem 5

Let SS be a finite set of points in space, and let Sx,Sy,SzS_x,S_y,S_z be its orthogonal projections onto the yzyz-, zxzx-, and xyxy-planes. Prove that ∣S∣2≤∣Sx∣∣Sy∣∣Sz∣|S|^2\le |S_x||S_y||S_z|.
Step 1 of 4: Start with one height
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In plain words

Start with one height

∣S∣=∣Sz∣,∣S∣≤∣Sx∣∣Sy∣|S|=|S_z|,\qquad |S|\le |S_x||S_y|
Detailed analysis

If all points have the same z-coordinate, identify that plane with its projection, so S=SzS=S_z. Each point of S_y has at most |S_x| preimages, hence ∣S∣≤∣Sx∣∣Sy∣|S|\le |S_x||S_y|; since ∣Sz∣=∣S∣|S_z|=|S|, the claim follows.