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 4 of 4: Combine the two terms
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In plain words

Combine the two terms

ab+cd≤(a+c)(b+d)\sqrt{ab}+\sqrt{cd}\le\sqrt{(a+c)(b+d)}
Detailed analysis

Apply the displayed inequality with a=∣Tx∣a=|T_x|, b=∣Ty∣b=|T_y|, c=∣Ux∣c=|U_x|, and d=∣Uy∣d=|U_y|. The projection equalities give ∣S∣≤(∣Sx∣∣Sy∣∣Sz∣)1/2|S|\le(|S_x||S_y||S_z|)^{1/2}; squaring proves the claim.