The Generalized Busemann-Petty Problem in Dimensions 2 and 3

If origin-symmetric convex bodies $K, L \subset \mathbb{R}^n$ satisfy $\mathrm{vol}_m(K \cap E) \leq \mathrm{vol}_m(L \cap E)$ for every $m$-dimensional subspace $E$ with $1 < m < n$, does $\mathrm{vol}_n(K) \leq \mathrm{vol}_n(L)$ follow? Answered affirmatively for subspace dimensions $m = 2$ and $m = 3$.

Result

Proved(see note)

Status

Partial result

AI contribution

AI-assisted

Method

Argument

Field

Convex geometry

Posed by

Herbert Busemann, Clinton Petty (hyperplane case); generalized form standard since

Year posed

1956

Years open

70y

Solved

2026-08-06

Model

ChatGPT 5.6 Sol

Vendor

OpenAI

Collaborators

Cheng Lin, Yu-De Liu, Ge Xiong

Verification

Unreviewed

Publication

Preprint

Significance

30 / 100*

Disclosed cost

—

Wikipedia

No dedicated article