Modified Shephards problem on projections of convex bodies

We disprove a conjecture of A. Koldobsky asking whether it is enough to compare $(n-2)$-derivatives of the projection functions of two symmetric convex bodies in the Shephard problem in order to get a positive answer in all dimensions.

💡 Research Summary

The paper addresses a long‑standing question in convex geometry concerning the Shephard problem, which asks whether a pointwise comparison of projection volumes of two convex bodies implies a comparison of their total volumes. In dimensions two the answer is affirmative, but in dimensions three and higher counterexamples are known. Alexander Koldobsky conjectured that the failure in higher dimensions might be remedied by comparing higher‑order derivatives of the projection functions: specifically, if the (n‑2)‑th derivatives of the projection functions of two origin‑symmetric convex bodies coincide (or one dominates the other) in every direction, then the bodies should have the same ordering of volumes. This conjecture, if true, would provide a powerful analytic criterion based on Fourier methods and the theory of k‑intersection bodies.

The authors disprove Koldobsky’s conjecture in all dimensions n ≥ 4. Their approach combines harmonic analysis on the sphere with a careful construction of convex bodies whose spherical harmonic expansions agree up to degree n‑2 but diverge in higher degrees. They begin by recalling that the projection function (P_K(\theta)=|K|_{\theta^\perp}) can be expressed via the Fourier transform of the support function, and that applying the spherical Laplacian (\Delta) repeatedly isolates higher‑degree spherical harmonic components. Consequently, the (n‑2)‑th derivative (\Delta^{(n-2)/2}P_K) depends only on the harmonic coefficients of degree ≤ n‑2.

Exploiting this observation, the authors select two families of symmetric convex bodies (K) and (L) whose harmonic coefficients satisfy \