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The Generalized Busemann-Petty Problem in Dimensions 2 and 3

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generalized-busemann-petty-2-3Geometry & topologyposed by Herbert Busemann, Clinton Petty (hyperplane case); generalized form standard since, 1956recorded: partial

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Statement

If origin-symmetric convex bodies K,L⊂RnK, L \subset \mathbb{R}^n satisfy volm(K∩E)≤volm(L∩E)\mathrm{vol}_m(K \cap E) \leq \mathrm{vol}_m(L \cap E) for every mm-dimensional subspace EE with 1<m<n1 < m < n, does voln(K)≤voln(L)\mathrm{vol}_n(K) \leq \mathrm{vol}_n(L) follow? Answered affirmatively for subspace dimensions m=2m = 2 and m=3m = 3.

Context

Settles subspace dimensions 2 and 3; the generalized problem stays open for larger m.

A direct descendant of the Busemann-Petty problem, one of the best-known questions of convex geometry, whose hyperplane case took four decades and several landmark papers to settle.

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  • #1

    Attempt 1

    proof attemptChatGPT 5.6 Sol with Cheng Lin, Yu-De Liu, Ge Xiong ·
    AI involvement
    ai assisted
    — a person led the work and used a model along the way.
    models
    ChatGPT 5.6 Sol
    people
    Cheng Lin, Yu-De Liu, Ge Xiong

    The declaration in full: "Theorem 2.3 was found with the help of ChatGPT 5.6 Sol, which led us to prove Theorem 3.2." One named theorem, which unlocked the paper's crucial representation formula.

    Settles subspace dimensions 2 and 3; the generalized problem stays open for larger m.

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