Why is Volume^2 at most product of the 3 projections?

Is there a simple proof for
\text{Vol}^2(P)\le \prod_{i=x,y,z} \text{Area}(\text{Proj}_i(P)),
where $P\subset \mathbb R^3$ and $\text{Proj}_z(P)$ denotes the projection of $P$ to the $z=0$ plane?

I have a complicated proof for this (in any dimension) using shifts but I wonder if this inequality is something well-known/simple.

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This is a special case of the Loomis-Whitney inequality. First, some biographical information:

  • Lynn Harold Loomis was an analyst. He got his PhD under Salomon Bochner at Harvard in 1942. Later he would teach at the same school. He co-aurthored Advanced Calculus with Shlomo Sternberg.
  • Hassler Whitney was a reknown topologist. He came from a rather learned family in New York City, studied at Yale where he got degrees in Physics and Music.

The Loomis Inequality appears in An Inequality Related to the Isoperimetric Inequality. The result states:

Let $m$ be the measure of an open subset of Euclidean $n$-space, $O \in \mathbb{R}^n$. Let $m_1, \dots, m_n$ be the $(n-1)$-dimensional measures of the projections on the coordinate hyperplanes. Then $m^{n-1} \leq m_1\dots m_n$

We can take $n=3$. The Loomis-Whitney paper is 2 pages long and we will summarize it here:

  • Step 1 Reduce to a counting problem:

    Let $S$ be a set of cubes from the cubical subdivision of $n$-space and let $S_i$ be the set of $n$-cubes by projecting the cubes of $S$ onto the $i$th coordinate hyperplane. Let $N$ be the number of cubes in $S$ and $N_i$ be the number of cubes in $N_i$ then $N^{n-1} \leq N_1 \dots N_n$

  • This cubic apprxoximation works since the number of cubes is a good approximation to the volume or “measure” of the set $S$. We just set the cubes of size $\delta$ and count.

  • Step 2 How do we prove the above lemma? The proof is by induction on the number of dimensions:

    • The cubes project into intervals $I_1, \dots, I_k$ if we project to the first axis. Here $k$ might be very large set of intervals.
    • For each interval $I_i$, let $T_i$ be the set of cubes projecting into $T_i$. Let $T_{i \to j}$ be the projection of $T_i$ onto the $j$-th hyperplane.
    • Let $a_i$ be the number of cubes in $T_i$. We can count the cubes so that
      $$ \begin{align} \sum_{i=1}^k a_i = N \\
      \sum_{i=1}^k a_{i \to j} = N_j \end{align}$$
      and by induction hypothesis $a_i^{n-2} \leq a_{i \to 2} \dots a_{i \to n}$ which implies $a_i^{n-1} \leq N_i a_{i \to 2} \dots a_{i \to n}$
    • The last step follows from Hölder’s inequality …

As I have learned from Gabor Tardos (and Peter Csikvari) this is well-known and can be proved easily using the submodularity of entropy. First we reduce the problem to the finite case by approximating $P$ by a collection of cubes. Then let us select a cube uniformly random and denote its $i$-th coordinate by $X_i$. We have
$$\log Vol^2(P)=H(X_1,X_2,X_3)+H(X_1,X_2,X_3)\le H(X_1,X_2)+H(X_1,X_3)-H(X_1)+H(X_1,X_2,X_3)\le H(X_1,X_2)+H(X_1,X_3)+H(X_2,X_3)\le \Pi_{i=x,y,z} Area(Proj_i(P)).$$