The CUBOID Inequalities

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We start with a general cuboid with dimensions L, B and H (for Length, Breadth and Height) and introduce the letters P, S and V for Perimeter, Surface area and Volume where P=4(L+B+H), S=2(LB+BH+LH), and V=LBH. We will use the AGM (Arithmetic-Geometric Mean) inequality a number of times in the following.

First, using the AGM Inequality on L+B+H, we get L + B + H≥3(LBH)1/3 with equality iff L=B=H. This is equivalent to (P/12)3≥V.

Next, using the AGM Inequality on LB+BH+LH, we get LB+BH+LH≥3(L2B2H2)1/3 with equality iff LB=BH=LH iff L=B=H.

This is equivalent to (S/6)3≥V2.

If we combine these inequalities by multiplying both left sides and both right sides we get PS/72≥V and it is easy to show that this is also with equality iff L=B=H.

Now, (S/2)2=L2B2+B2H2+L2H2+2(L+B+H)V=L2B2+B2H2+L2H2+PV/2.

Also, (LB-BH)2+(BH-LH)2+(LH-LB)2≥0 with equality iff L=B=H. Multiplying this out gives 2(L2B2+B2H2+L2H2)≥PV/2.

Hence, (S/2)2≥PV/4+PV/2=3PV/4.

From the above we get S2≥3PV and yet again with equality iff L=B=H.

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