Convex and metric geometry·OpenAI Math·Result 088·Lean formalisation listed
Sharp projection-body inequalities and a counterexample to simplex maximization
One of the 372 results in OpenAI's public mathematics catalogue, filed under convex and metric geometry: 2 preprints, the latest dated 24 September 2026. This page quotes OpenAI's own summary of the claim, links the papers and abstracts, and embeds the animation.
In one line: OpenAI reports a result titled ‘Sharp projection-body inequalities and a counterexample to simplex maximization’ (convex and metric geometry) in 2 preprints, the latest dated 24 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More, is on YouTube.
Chapter 56 of 83 in Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · starts at 19:05 · watch from there on YouTube →
What OpenAI claims
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. (Lean)
Papers (2)
Petty’s projection-volume conjecture in dimensions at least four
OpenAI's abstract 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.
A product counterexample to the simplex maximum for projection-body volume
OpenAI's abstract 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.
Status
Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted 2 preprints, the latest dated 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- 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. (Lean)
- Where is the proof?
- In the paper, Petty’s projection-volume conjecture in dimensions at least four (PDF).
- Is there a video?
- Yes, on YouTube (28:48, chapter 56).
Also in
The video
Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · Compilation · 28:48 · chapter 56 from 19:05 · description, sources and chapters →
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