Rigidity and a sharp halfspace benchmark for Gaussian cylinders

Definitions and mathematical statement

Let $\gamma_n$ be standard Gaussian probability on ${\mathbb{R}}^n$, let $K$ be measurable and convex with $\gamma_n(K)>0$, let $X\sim\gamma_n(\,\cdot\mid K)$, and let $\theta$ be a unit vector. A cylinder in direction $\theta$ has the form $C=E+{\mathbb{R}}\theta$, $E\subset\theta^\perp$. The source asks whether \[ \operatorname{Var}\langle X,\theta\rangle=1 \] forces such a cylinder, and whether deficit $\varepsilon=1-\operatorname{Var}\langle X,\theta\rangle$ forces small symmetric difference from one. The extracted control character is read as $\varepsilon$. Standard covariance and unit direction are explicit normalization assumptions; the original does not specify them or a measure for symmetric difference.

The source remarks state $\operatorname{Var}\langle X,\theta\rangle\le1$, obtainable for example by Brascamp--Lieb, and refer to Sidak's lemma in the symmetric case. The supplied aid attributes exact rigidity to Vempala's 2010 normal-subspace theorem: unit conditional variance is equivalent to invariance in direction $\theta$. Literal set invariance requires an appropriate closed convex representative; for arbitrary measurable convex sets, boundary changes invisible to Gaussian conditioning require an almost-everywhere or closure interpretation. For closed full-dimensional $K$, the claimed conclusion is $K=E+{\mathbb{R}}\theta$. The full theorem hypotheses are not supplied.

For its quantitative benchmark, define \[ \delta_\theta(K)=\inf_{E\subset\theta^\perp\ \mathrm{measurable}} \gamma_n\left(K\mathbin{\triangle}(E+{\mathbb{R}}\theta)\right). \] This is absolute standard-Gaussian distance, not Lebesgue or conditional-relative distance. For $n\ge2$, take the centered halfspace $K=\{x:\langle u,x\rangle\ge0\}$, where $u$ is unit, and put $\rho=\langle u,\theta\rangle$. The aid reports \[ \gamma_n(K)=1/2,\qquad \varepsilon=\frac2\pi\rho^2,\qquad \delta_\theta(K)=\frac{\arcsin|\rho|}{\pi} =\frac{\arcsin\sqrt{\pi\varepsilon/2}}{\pi}. \] Here $0\le\varepsilon\le2/\pi$. For $|\rho|<1$, the optimum is attained by the convex cylinder $\{x:\langle u-\rho\theta,x\rangle\ge0\}$. At $|\rho|=1$, every cylinder has error $1/2$, and whole space or the empty cylinder attains it; this avoids treating the zero projected normal as a genuine halfspace normal.

As $\varepsilon\downarrow0$, the reported distance is asymptotic to $\sqrt{\varepsilon/(2\pi)}$. Thus no uniform estimate $\delta_\theta(K)\le C\varepsilon^\beta$, with a fixed finite $C$, can hold for this family with $\beta>1/2$. The benchmark fixes Gaussian measure at one half, so it is not an artifact of vanishing body probability. It supplies a necessary exponent limit, not an upper stability theorem for all convex sets. The original quantitative question still needs its distance convention, dimensional/probability dependence, and regularity assumptions specified.

Short English statement

The source asks whether maximal conditional Gaussian variance forces a cylinder and whether near-maximal variance gives quantitative closeness. The supplied aid reports exact rigidity and computes tilted halfspaces showing that a uniform Gaussian-distance stability exponent cannot exceed one half.

Sources

• Local UnsolvedMath registry, permanent ID $20001291$, source code catalogue entry; aim-convex-geometry-notes.json, notes[22], tag problem.
• AIM workshop "Algorithmic Convex Geometry," 5--9 November 2007, Problem 4, Ryan O'Donnell (printed "O'Donnnell"): https://aimath.org/WWN/convexgeometry/convexgeometry.pdf. Statement status reconstructed_unverified. No PDF was accessed for this draft.
• Source references Sidak's lemma and the Brascamp--Lieb inequality without full citations. Vempala's 2010 normal-subspace theorem is named by the aid without paper title, theorem number, or URL. The halfspace benchmark has no separate authoritative citation.
• Research attempt 1: partial_result, status partially_solved, classification PARTIAL-PROGRESS, rigor label proved, novelty confidence low. Created and updated 14 August 2026. This machine-generated aid is not peer reviewed; independently verify before citation.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0.