geometry-topology / Convex geometry
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 $m = 3$.
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Append-only history
Settles subspace dimensions 2 and 3; the generalized problem stays open for larger m.
Research memory
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 $m = 3$.
Settles subspace dimensions 2 and 3; the generalized problem stays open for larger m.
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