This theorem, nowadays proved as a consequence of the Brunn-Minkowski inequality (which see for proof!), sounds like an innocuous statement on volumes in convex bodies. The same cannot be said for the proof. Modern proofs are easy but require the Brunn-Minkowski inequality, itself nontrivial.

This diagram shows a 2-dimensional case of Brunn's inequality. You'll need to have <pre> tags displayed in a monospaced font, or you won't see anything. We cut the convex body C along 3 parallel "hyperplanes" (in this case they're just lines) x,y,z. The lengths satisfy l(y) > l(x) = min(l(x),l(z)), as required by the inequality.

    _,-"":       :""""---...__
  ,' 	 :       :            `.
 / :	 :       :              `.
|  :     :       :         C      \
|  :x    :       :      	   |
|  :     :y      :      	    |
 | :     :       :z     	     |
  \:     :       :      	     |
   `.	 :       :      	     |
     `.	 :       :      	    |
       `._       :      	   /
	  `._    :             __,'

Theorem. Let C⊆Rd be a convex body; let X, Y and Z be three parallel (d-1)-dimensional hyperplanes. The intersections C∩X, C∩Y and C∩Z are (d-1)-dimensional convex bodies; let their (d-1)-dimensional volumes be
VX = Vd-1(C∩X)
VY = Vd-1(C∩Y)
VZ = Vd-1(C∩Z)
Then VYmin(VX,VZ).

Pick a direction (a vector of length 1) v. Let Xt be the hyperplane orthogonal to v, at distance t from the origin. Brunn's inequality guarantees us that the function

V(t) = Vd-1(C∩Xt)
is unimodular: it is initially monotone ascending, reaches its only local maximum (which is therefore its global maximum), and is then monotone descending.

You can see a proof on the Brunn-Minkowski inequality node (which uses that inequality).


A boo-boo kitty offers the following explanation of what the theorem is all about. Suppose you are given a tomato (or, as we mathematicians would explain in simpler language, "let T⊂R3 be a tomato"). Cut 3 parallel cuts through the tomato, and look at the face of each cut. Then the face with the smallest area is one of those created by the two end cuts, not the face created by the middle cut.

Or, in other words, if you cut 3 slices from a tomato, the smallest slice won't be the middle one...

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