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 2 of 4: Split into two nonempty height ranges
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In plain words

Split into two nonempty height ranges

∣S∣=∣T∣+∣U∣,∣Sx∣=∣Tx∣+∣Ux∣,∣Sy∣=∣Ty∣+∣Uy∣|S|=|T|+|U|,\qquad |S_x|=|T_x|+|U_x|,\qquad |S_y|=|T_y|+|U_y|
Detailed analysis

Choose a horizontal plane splitting S into nonempty parts T,U. Their yz- and zx-projections are disjoint by height, while Tz,Uz⊆SzT_z,U_z\subseteq S_z, so the displayed projection equalities hold.