Problem 5

Let

S

be a finite set of points in space, and let

S_x,S_y,S_z

be its orthogonal projections onto the

yz

zx

-, and

xy

-planes. Prove that

|S|^2\le |S_x||S_y||S_z|

Step 3 of 4: Apply the induction hypothesis

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In plain words

Apply the induction hypothesis

|S|\le |T_x|^{1/2}|T_y|^{1/2}|T_z|^{1/2}+|U_x|^{1/2}|U_y|^{1/2}|U_z|^{1/2}

Detailed analysis

By induction, $|T|\le(|T_x||T_y||T_z|)^{1/2}$ and the analogous bound holds for U. Adding them and using $|T_z|,|U_z|\le |S_z|$ gives the displayed estimate.