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The Generalized Busemann-Petty Problem in Dimensions 2 and 3

Geometry & topology · posed by Herbert Busemann, Clinton Petty (hyperplane case); generalized form standard since, 1956 · 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.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Cheng Lin, Yu-De Liu and Ge Xiong, using ChatGPT 5.6 Sol

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai assisted
    — a person led the work and used a model along the way.

    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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