Closedness, dimensional correction, and a dual certificate for polar-zonoid intersection bodies

Definitions and mathematical statement

For a fixed dimension $n\geq3$, let $\mathcal K_s^n$ be the space of full-dimensional origin-symmetric convex bodies in ${\mathbb{R}}^n$, with Hausdorff distance. For $K\in\mathcal K_s^n$, its intersection body $IK$ has radial function \[ \rho_{IK}(u)=\operatorname{vol}_{n-1}(K\cap u^\perp),\qquad u\in S^{n-1}. \] For an origin-containing convex body $L$, its polar is $L^\circ=\{y:\langle x,y\rangle\leq1\ \forall x\in L\}$. A zonoid is a Hausdorff limit of finite Minkowski sums of line segments. Here $IK$ is convex in the origin-symmetric convex setting, so saying that $IK$ is the polar of a zonoid is equivalent to saying that $(IK)^\circ$ is a zonoid.

R. Schneider's requested Baire-category assertion is that \[ \{K\in\mathcal K_s^n:(IK)^\circ\text{ is not a zonoid}\} \quad\text{is residual in }\mathcal K_s^n, \] where residual means that its complement is meager (a countable union of nowhere dense sets). The source says "most" in this sense, not in a probability or volume sense. Its excerpt does not specify a dimension or topology; the fixed-$n\geq3$ Hausdorff formulation is the dimensionally qualified version supplied by the research aid. In dimension two the aid says the literal assertion is false: $IK$ is a rotated, scaled copy of $K$ and every origin-symmetric planar convex body is a zonoid.

The supplied aid reports that the exceptional set \[ E_n=\{K\in\mathcal K_s^n:(IK)^\circ\text{ is a zonoid}\} \] is closed. Thus the proposed generic assertion reduces to density of its open complement, equivalently to $E_n$ having empty interior. The aid also describes an affine-invariant version on the Banach--Mazur compactum of linear-equivalence classes; this does not replace the stated Hausdorff formulation with an unspecified new topology.

For completeness, the aid's dual obstruction can be stated with explicit integral normalizations. For $\varphi$ on $S^{n-1}$ define the unnormalized spherical Radon transform by \[ (R\varphi)(u)=\int_{S^{n-1}\cap u^\perp}\varphi(v)\,d\sigma_{n-2}(v). \] Then its notation is \[ f_K(u)=\frac{n-1}{R(\rho_K^{n-1})(u)}=h_{(IK)^\circ}(u), \] where $h_L(u)=\sup_{x\in L}\langle x,u\rangle$ is the support function. For a finite even signed Borel measure $\nu$, even meaning $\nu(A)=\nu(-A)$, write \[ (C\nu)(u)=\int_{S^{n-1}}|\langle u,v\rangle|\,d\nu(v). \] The reported certificate is \[ (IK)^\circ\text{ is not a zonoid} \quad\Longleftrightarrow\quad \exists\nu:\ C\nu\geq0\text{ pointwise},\quad \int_{S^{n-1}}f_K\,d\nu<0. \] The aid states that a strict witness persists under sufficiently small Hausdorff perturbations. These are attributed partial results; closedness and a certificate for a given example do not themselves establish the required density.

Short English statement

Schneider asks whether a typical symmetric convex body's intersection body fails to be the polar of a zonoid, with typical understood in the Baire-category sense. The supplied partial work makes the exceptional class closed and gives a strict separating-measure certificate, leaving density as the missing step. Dimension two is excluded because the literal assertion fails there.

Sources

• Local UnsolvedMath registry, permanent ID 20001306, exact source code catalogue entry; canonical aim-convex-geometry-notes.json, notes[37], source item 0, tag "problem", workshop Fourier analytic methods in convex geometry. Original attribution: R. Schneider. Supplied URL: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf.
• The original states the Baire-category meaning of "most" but supplies no separate bibliographic references. Its trailing isolated "1" is retained as extraction evidence rather than a mathematical condition. The verification consists only of "The canonical record reads"; original remarks and literature field are empty.
• Supplied research metadata: PARTIAL-PROGRESS, registry status "partially_solved", statement status "exact"; attempt 1, result "partial_result", problem status at run "open", recorded rigor "proved", novelty confidence "low", candidate type "dual obstruction lemma". The research aid explicitly is machine generated, not peer reviewed, and requires independent verification before citation. No outside source was visited.