# OWR-12697711-006: discotope irreducibility

Frozen corpus: ulamai/UnsolvedMath v1.6.0, commit
`c6e7f41b4eb6ce876feb83e7eb626be1b243fd7e`, record 30005473.
The frozen statement has interleaved extraction from two different conjectures.
Only Conjecture 1 on discotopes is this record's target. The following rank-one
convexity/PDE conjecture, the zonoid characterization problem, and degree or
birationality questions are not included.

Primary source: Chiara Meroni, *Two convex conjectures for different flavours*,
in *New Directions in Real Algebraic Geometry*, Oberwolfach Report 15/2023,
pp. 829-832, DOI 10.4171/OWR/2023/15. Definition 1 and Conjecture 1 are on
printed p. 830 (PDF page 18). Visually verified from the preserved PDF.

Let D_i be linear images of Euclidean closed unit balls in R^d, with intrinsic
dimension m_i >= 2. Let D be their Minkowski sum. Let E be the complex Zariski
closure of the exposed points of D. The conjecture says E is irreducible for
generic discs. An exposed point means a singleton supporting face, not an
arbitrary point lying on a supporting hyperplane.

Earlier formulation: Fulvio Gesmundo and Chiara Meroni, *The Geometry of
Discotopes*, Le Matematiche 77(1) (2022), 143-171, DOI 10.4418/2022.77.1.8.
Definition 3.3 (p. 149) defines S as the complex Zariski closure of
`(sum_i relative_boundary(D_i)) intersect boundary(D)`. Conjecture 8.2 (p. 167)
says S is irreducible for a generic discotope without segment summands. These
definitions are not identified without a proof. The paper assumes D spans its
ambient space (p. 147); disc boundaries are relative to their spans (p. 146).
Genericity is with respect to the full-rank basis matrices (p. 146).

The proposed result proves E irreducible even without genericity, and proves
S=E under an explicit nonempty Zariski-open condition on all subcollections of
disc spans. Thus both precise conjectures, but no adjacent question, are covered.


# Analytic support parametrization and generic face approximation

Date: 2026-10-27. Originating researcher: primary Codex lane for Alper Ferudun.
This is a written proof, not a formal proof-assistant certificate.

## Setup

Write D_i=A_i B_{m_i}, where A_i is a real d by m_i matrix of full column
rank, B_m is the closed Euclidean unit ball, and m_i>=2. Set Q_i=A_i A_i^T
and L_i=im(A_i). For S work in the span of all L_i, hence assume D is full
dimensional. Define

U=R^d minus union_i ker(A_i^T),
p_i(u)=A_i A_i^T u / ||A_i^T u||, and F(u)=sum_i p_i(u).

All norms and square roots in this formula are real and positive. No complex
branch or simply connected parameter space is used.

## Lemma 1: U is nonempty and path connected

Each removed subspace K_i has codimension m_i>=2. A finite union of proper
linear subspaces cannot cover R^d. For a,b in U, choose c outside every
K_i+R a and K_i+R b. These spaces have dimension at most d-1. The segment
[a,c] avoids K_i: an intersection (1-t)a+t c in K_i, with t>0, would put
c in K_i+R a, while t=0 is excluded by a in U. The same holds for [b,c].
This gives a polygonal path in U. Also U is open.

## Lemma 2: Exp(D)=F(U)

For any u, maximizing <u,A_i z> over ||z||<=1 is maximizing <A_i^T u,z>.
If A_i^T u is nonzero, Cauchy-Schwarz gives the unique maximizer
z=A_i^T u/||A_i^T u||, hence p_i(u). If it is zero, the whole disc is the
maximizing face. For a Minkowski sum, the face exposed by u is the sum of the
individual faces: the sum of their nonnegative support deficits vanishes
exactly when each deficit vanishes. This face is a singleton exactly when
every summand face is a singleton. Indeed, fixing points in the other summands
shows a non-singleton face cannot collapse under Minkowski addition. This
establishes both inclusions, including all exceptional normal directions.

## Lemma 3: a connected real-analytic image has irreducible complex closure

Let V be a nonempty connected open subset of R^d and f:V->R^n be real
analytic. Suppose complex polynomials P,Q satisfy (P Q)(f(u))=0 for all u.
If P composed with f is not identically zero, it is nonzero on a nonempty
open subset. Q composed with f vanishes there and hence everywhere by the
identity theorem for real-analytic functions (applied to real and imaginary
parts). Thus the vanishing ideal of f(V) in C[x_1,...,x_n] is prime. Its
complex Zariski closure is irreducible. Continuity alone would not suffice.

## Theorem 1: unconditional irreducibility of E

On U every quadratic form u^T Q_i u is strictly positive. The positive square
root and its reciprocal are real analytic there. F is therefore real analytic.
Lemmas 1-3 prove that the complex Zariski closure E of Exp(D) is irreducible.
No transversality, common intrinsic dimension, full-dimensional summand,
dimension inequality, or generic metric is needed.

## The generic span condition

Impose, for every subset J of {1,...,N},

dim(sum_{i in J} L_i)=min(d, sum_{i in J} m_i).                 (GP)

This condition holds on a nonempty Zariski-open set of the matrix parameter
space. Each failure is a maximal-rank failure of a concatenated block matrix,
cut out by minors; there are only finitely many J. Nonemptiness follows by
taking all columns from a moment curve (1,t,...,t^{d-1}) with distinct real t.
Every set of at most d such columns is independent by the Vandermonde
determinant. This construction also ensures each A_i has full column rank.

## Theorem 2: under (GP), S=E

Let X=(sum_i relative_boundary(D_i)) intersect boundary(D). Since each
exposed point is a sum of unique boundary support points, Exp(D) is contained
in X. It remains to show X is contained in the Euclidean closure of Exp(D).

Take x=sum_i A_i z_i in X with ||z_i||=1, and a nonzero support normal u
to D at x. Such a normal exists for a boundary point of a full-dimensional
compact convex body. Equality in the sum of support inequalities implies
A_i z_i lies in the face of D_i exposed by u. Put J={i:A_i^T u=0}.
For i not in J, A_i z_i=p_i(u). If J is empty, x=F(u) already.

Otherwise sum_{i in J} L_i is contained in u-perp and has dimension <=d-1.
By (GP), sum_{i in J} m_i<=d-1 and these subspaces form a direct sum.
Consequently the stacked map v -> (A_i^T v)_{i in J} has full row rank and is
surjective. Choose v with A_i^T v=z_i for every i in J. For positive epsilon,
A_i^T(u+epsilon v)=epsilon z_i for i in J, so p_i(u+epsilon v)=A_i z_i
exactly. For i not in J, these projections stay nonzero for every sufficiently
small epsilon, and p_i(u+epsilon v) tends to p_i(u). Thus u+epsilon v belongs
to U and F(u+epsilon v) tends to x. The positivity of epsilon matters.

Every polynomial vanishing on Exp(D) also vanishes on its Euclidean closure,
hence on X. Conversely Exp(D) is contained in X. Their complex Zariski
closures are equal. Therefore S=E and S is irreducible by Theorem 1.

## Scope controls

1. Segments cannot be allowed in Theorem 1. For the sum of the planar unit
   disc and [-e_1,e_1], exposed points form the two outer open semicircles.
   Their Zariski closure is the union of two distinct translated circles.
2. S=E is not unconditional for a fixed summand representation. Let D_1=D_2
   be the unit disc in the xy-plane and D_3 the unit disc in the xz-plane.
   The point (0,0,1) is a sum of three disc-boundary points and lies in the
   supporting face with normal e_3, hence belongs to X. Exposed points are
   sums of a point (a,y,0) on the radius-two xy-circle and a point (b,0,z)
   on the unit xz-circle. They satisfy
   G(x,y,z)=(x^2+y^2-z^2-3)^2-4*x^2*(1-z^2)=0,
   by eliminating a+b=x, a^2+y^2=4, b^2+z^2=1. But G(0,0,1)=16.
   Thus (0,0,1) is not in E. Here (GP) fails for {1,2}.
3. The entire critical locus of the complex addition map may contain other
   components. Its irreducibility, birationality, and degree are not asserted.
4. The general zonoid recognition and rank-one convex hull questions in the
   report remain separate. This proof addresses the exact discotope conjecture.


# Adversarial self-audit

2026-10-27; primary lane. Independent review: false. Formal verification: false.

- **Connected versus irreducible:** connectedness alone is insufficient. The
  proof uses analytic compositions of polynomials and the identity theorem to
  prove the vanishing ideal is prime over C. No assertion about arbitrary
  connected images is used.
- **Real versus complex:** the parameter domain is real and open. Complex
  polynomials have complex-valued real-analytic compositions; the real and
  imaginary parts obey the identity theorem. The ideal is explicitly in C[x].
- **Square-root branches:** every denominator is a positive real square root
  on U. There is no attempted global holomorphic square root on a complex
  discriminant complement and no unjustified monodromy argument.
- **Exceptional directions:** if any projected normal vanishes, the exposed
  face contains a translate of an entire positive-dimensional disc. Such a
  normal cannot expose a singleton. Hence F(U) is exactly Exp(D).
- **Singularities of the image:** F need not be injective or an immersion.
  The analytic-ideal argument requires neither property.
- **Non-round ellipsoids:** the support numerator is A A^T u, not the
  orthogonal projection onto the span. The denominator is ||A^T u||.
- **Boundary conventions:** each disc uses its relative boundary; D uses
  the boundary in its total span. The older paper makes the same convention.
- **Equality S=E:** not assumed. A boundary support normal reduces the problem
  to a simultaneous linear prescription on the annihilated summands. (GP)
  gives full row rank. All other projected normals remain nonzero near u.
- **Size of the exceptional family:** u is nonzero, so a family contained in
  u-perp cannot span R^d. (GP) then forces sum m_i<=d-1; the required
  prescriptions are compatible. This is the decisive use of genericity.
- **Genericity is nonvacuous:** finite block-rank conditions are Zariski open,
  and distinct moment-curve columns provide one real point satisfying all of
  them. No claim is made that every nongeneric family is covered by S=E.
- **Insufficient closure notions:** inclusions in Euclidean closure imply
  inclusions in complex Zariski closure because complex polynomials are
  continuous on real points. No equality of Euclidean and Zariski topology is
  used.
- **N=1 and full-dimensional discs:** U is punctured R^d for a full-dimensional
  summand, still connected for d>=2. If a full-dimensional summand occurs,
  it cannot be one of the annihilated summands for a nonzero normal.
- **Codimension-one failure:** the stadium counterexample shows why segment
  summands cannot simply be included. A second exact control shows S!=E for a
  nongeneric representation, even though E remains irreducible.
- **Novelty:** the support-face description is already in Gesmundo-Meroni
  Section 3 and must be credited. The identity-theorem argument is elementary
  standard analysis. The claimed contribution is their application to general
  E and the explicit generic face-approximation argument establishing S=E.
  A bounded current search is not a certificate of absolute priority.

Mathematical assessment: the two theorems above have no outstanding proof
obligation after this audit. Exact-source coverage: both OWR Conjecture 1 and
Gesmundo-Meroni Conjecture 8.2, with the stated conventions. Computational
fixtures are additional regression evidence, not the proof of either theorem.


# Source and bounded novelty review

Checked 2026-10-27. No absolute priority or exhaustive literature certification.

The source theorem is Conjecture 1, p. 830, in Chiara Meroni's contribution
*Two convex conjectures for different flavours* (pp. 829-832), OWR 15/2023,
DOI 10.4171/OWR/2023/15. The underlying paper is Gesmundo-Meroni,
*The Geometry of Discotopes*, Le Matematiche 77(1) (2022), 143-171,
DOI 10.4418/2022.77.1.8; arXiv:2111.01241v2 (28 July 2022).
The published and arXiv PDFs were retrieved separately and hashed.

Decisive source locations, read in context and visually inspected:

- Published pp. 146-147: basis-matrix definition of ellipsoidal discs, full
  dimensionality in the total span, genericity in basis-matrix parameters.
- Published pp. 147-149: the support-normal face description and Definition
  3.3 of S. The paragraph following Definition 3.3 explicitly notes that S
  is not a priori the exposed-point closure.
- Published p. 167: Conjecture 8.2 and the separate expectations about the
  full critical locus, birationality, and degree.
- OWR p. 830: the E formulation, its genericity convention and Conjecture 1.

The earlier results prove cases using joins of quadrics and critical loci;
all two-dimensional summands and all full-dimensional summands are covered
there. Those special cases and the support-face formula are not claimed as
new here. The elementary analytic-image lemma is a standard consequence of
the real-analytic identity theorem, supplied with a proof. The contribution
claimed is its application to arbitrary disc dimensions >=2 and the explicit
generic normal perturbation identifying S and E.

Targeted searches included the exact problem ID, discotopes with conjecture
8.2, irreducibility, exposed points, support maps, analytic maps, proof, and
2024-2026, plus Zenodo/EvidencePress/EulerSolve queries. Search records are
preserved in literature_search.json. The journal record, arXiv version
history, and Chiara Meroni's current research list were read:

- https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/2338
- https://arxiv.org/abs/2111.01241
- https://merochia.wixsite.com/chiara-meroni/research
- https://fulges.github.io/
- https://link.springer.com/article/10.1007/s00454-022-00451-3

The last source, Mathis-Meroni's *Fiber Convex Bodies*, concerns fiber bodies
and cites the discotope work; its Section 5.2 concerns discs in R^3 and is not
a proof of the general conjecture. Meroni's 2022 thesis is a further version
of the earlier work, not independent proof credit. The current author list
includes 2026 work but no subsequently listed solution to this conjecture.
Absence from a list or search result is not proof of absence from literature.

No identical general result was found within this bounded search. The source
remains listed as open in the frozen corpus; that label was not used as proof
of novelty. No first-ever or certified-priority assertion is made.

Local collision check at selection: no discotope manuscript in the primary
inventory, peer publication/audit logs, or paper sources. Fresh public HF and
peer reconciliation must precede publication; no duplicate notice is intended.