International Mathematics Competition
for University Students
2023

IMC2023: Day 2, Problem 9

Problem 9. We say that a real number \(\displaystyle V\) is good if there exist two closed convex subsets \(\displaystyle X\), \(\displaystyle Y\) of the unit cube in \(\displaystyle \mathbb{R}^3\), with volume \(\displaystyle V\) each, such that for each of the three coordinate planes (that is, the planes spanned by any two of the three coordinate axes), the projections of \(\displaystyle X\) and \(\displaystyle Y\) onto that plane are disjoint.

Find \(\displaystyle \sup\{V\mid V \text{ is good}\}\).

Josef Tkadlec and Arseniy Akopyan

Hint: The two bodies can be replaced by a pair symmetric to the midpoint of the cube.

  Solution