Polynomial ridge completeness and switching obstructions for the four AIM directions

Definitions and mathematical statement

The assigned extraction merges Problem 22, attributed to A. Daurat and R. Gardner, with the beginning of Problem 23. The main question fixes linear forms \[ \ell_1=x,\qquad\ell_2=y,\qquad\ell_3=2x+y,\qquad\ell_4=-x+2y \] and asks whether every planar convex body has an exact representation \[ K=\{(x,y):g_1(\ell_1(x,y))+g_2(\ell_2(x,y))+g_3(\ell_3(x,y))+g_4(\ell_4(x,y))\ge0\} \] with measurable real univariate functions $g_i$. A ridge $g(\langle x,v\rangle)$ is constant along $v^\perp$; rescaling the two nonunit listed vectors only changes $g$. The source notes that this four-direction collection determines all planar convex bodies by X-rays and no three-direction collection does. Its broader measurable-set uniqueness claim needs the usual null-set and finite-X-ray/integrability conventions. Ridge normals and integration directions are perpendicular; the displayed collection is closed under that rotation, so its full four-direction set is unchanged.

The attached aid states that every bivariate polynomial of total degree at most three is a sum of univariate ridge polynomials on these four forms, and every quadratic uses only the first three. For an ellipse, put $\xi=x-h$, $\eta=y-k$, and $Q=a\xi^2+2b\xi\eta+c\eta^2$ with $a,c>0$, $ac>b^2$. The explicit identity \[ Q=(a-2b)\xi^2+(c-b/2)\eta^2+(b/2)(2\xi+\eta)^2 \] gives a three-ridge threshold $1-Q\ge0$. The univariate coefficients need not all have one sign; measurability imposes no such restriction.

The aid's complementary exact obstruction is a finite signed measure $\sigma=\sum_jc_j\delta_{x_j}\ne0$ whose pushforward by every $\ell_i$ is zero. If all positive-coefficient locations lie in $K$ and all negative-coefficient locations lie outside it, no literal threshold representation above is possible: the summed ridge values have signed sum zero, incompatible with nonnegative values inside and strictly negative ones outside. A compatible $16$-term switching family is \[ \sigma=\mathop{*}_{i=1}^4(\delta_0-\delta_{w_i}),\qquad0\ne w_i\in\ker\ell_i, \] with generic lengths chosen so its sixteen subset-sum locations are distinct. The criterion concerns exact pointwise sets; a finite-point obstruction alone does not exclude a representation up to planar null sets. No convex counterexample is provided by the summary, and the general convex-body representation question remains open in its assessment.

The appended incomplete Problem 23 defines, for $a\in S^{n-1}$ and $t>0$, slabs $\{x:\|x\|_\infty\le1,\ |\langle x,a\rangle|\le t\}$ and sections $\{x:\|x\|_\infty\le1,\ \langle x,a\rangle=t\}$, with volumes $V(a,t)$ and $A(a,t)$, and $f_m=(1,\ldots,1,0,\ldots,0)/\sqrt m$. It quotes $1\le A(a,0)\le\sqrt2$, with a lower equality at $f_1$ and a truncated upper equality direction. For the printed cube $[-1,1]^n$, the actual coordinate section has area $2^{n-1}$, so the numerical claim requires normalized area $A/2^{n-1}$ or a unit-volume cube. This normalization conflict and the missing final index are left explicit rather than imported into the ridge question.

Short English statement

Represent convex bodies by a threshold of four fixed ridge functions. The supplied aid proves polynomial and ellipse cases and gives an exact finite switching obstruction, while keeping pointwise and almost-everywhere claims distinct. The merged cube-section fragment has a separate normalization problem.

Sources

• Local UnsolvedMath registry, permanent ID $20001312$, source code catalogue entry; canonical aim-convex-geometry-notes.json, notes[43].
• American Institute of Mathematics, Fourier analytic methods in convex geometry, recovered Problem 22 attributed to A. Daurat and R. Gardner, with appended beginning of Problem 23: https://aimath.org/WWN/fourierconvex/fourierconvex.pdf. Verification reports the concatenation and comma repairs in the original PDF; no PDF inspection is claimed here.
• The appended section names Hensley, Ball and Oleskiewicz--Pełczyński for section bounds but gives no complete citations. The main question supplies no literature reference.
• Attached research attempt 1: partial result, recorded run status open, rigor proved, novelty confidence low; registry status partially solved, classification PARTIAL-PROGRESS. Its polynomial and switching results are machine generated and not peer reviewed; independently verify before citation.