Petty's projection-volume conjecture and simplex counterexamples at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

A
D
V
E
R
T

I
S
E
M
E
N
T

Petty's projection-volume conjecture and simplex counterexamples

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

thinking... about 3 hours remaining

If this game doesn't work on your computer, go here for help. (Lean version available!)

expertly designed by an internal OpenAI model

Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	shapes, measuring stuff	Levels:	2
Category:	Convex and metric geometry	Lean version:	YES! ✔
Rate this game! 4.7 out of 5 (7,042 votes)

>>> How to Play <<<

Sharp projection-body inequalities and a counterexample to simplex maximization. Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension.

>>> Level Select <<<

LEVEL 1 Petty’s projection-volume conjecture in dimensions at least four

released 2026-09-24 | 4 theorems · 9 lemmas · 20 proofs · 10,677 words | PLAY LEVEL 1 » (pdf)

We prove that ellipsoids uniquely minimize the volume of the projection body among convex bodies of fixed volume in every dimension at least four. This proves Petty's projection-volume conjecture in these dimensions.

LEVEL 2 A product counterexample to the simplex maximum for projection-body volume

released 2026-09-24 | 1 theorem · 2 lemmas · 6 proofs · 2,318 words | PLAY LEVEL 2 » (pdf)

The product of two ten-dimensional simplices has larger normalized projection-body volume than a twenty-dimensional simplex. This gives a counterexample to Brannen's proposed simplex maximum.

More Convex and metric geometry Games!
Bounded-distortion $L_1$ embeddings of planar and bounded-treewidth graphs	Universal optimality of the triangular lattice HOT!	The logarithmic Brunn–Minkowski conjecture HOT!	The optimal order of convex-body covering density
Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures	Subpolynomial dimension reduction in $L_p$	Hyperbolicity cones without semidefinite lifts	The Gaussian propeller conjecture

COOLMAITH GAMES SPONSORS: (not real)
Big Al's USED GPUs	Epsilon & Delta LLP attorneys at limit	QED Bail Bonds	Hilbert's Hotel always a vacancy!

Cool Links: openai/math Lean Mathlib arXiv the real Coolmath Games

You are visitor number 000719

Thanks for visiting CoolmAIth Games
© 1997-2026 CoolmAIth Games. A parody. Not affiliated with Coolmath.com LLC or OpenAI.
Every proof here was written by an internal OpenAI model and lives at github.com/openai/math.
Best viewed in Netscape Navigator 4.0 at 800x600