Irreducibility of exposed-point varieties of discotopes
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Abstract
We prove that the complex Zariski closure of the exposed points of any finite Minkowski sum of ellipsoidal discs of dimension at least two is irreducible. No genericity or full-dimensionality assumption is needed. This proves the exposed-point irreducibility conjecture stated by Meroni in the 2023 Oberwolfach report. Under an explicit generic condition on the summand spans, we also prove that this closure equals the purely nonlinear part defined by Gesmundo and Meroni, thereby proving their 2022 irreducibility conjecture. The proof uses a connected real-analytic support-point parametrization. A nongeneric example shows why the two varieties must be distinguished.
The idea in plain English
A discotope is formed by adding centered ellipsoidal balls, which may lie in different linear subspaces. A point is exposed if one nonzero real linear functional has that point as its unique maximizer.
The paper proves that the complex algebraic closure of all exposed points is irreducible whenever every summand has dimension at least two. The conclusion covers arbitrary configurations, including repeated discs and sums lying in a smaller ambient span.
The earlier purely nonlinear part is a different algebraic set. The paper proves equality with the exposed-point closure under an explicit generic-position condition and gives a nongeneric example where equality fails.
The precise result
For every finite positive sum D=D_1+⋯+D_N⊂R^d of centered ellipsoidal balls whose actual span dimensions are all at least two, E(D)=Zar_C(Exp(D)) is irreducible over C. Exp(D) is the actual set of unique maximizers of nonzero real linear functionals. No genericity or full-dimensionality assumption is required.
Separately, suppose D is full-dimensional and every subset J of summand spans L_i satisfies dim(Σ_(i∈J)L_i)=min(d,Σ_(i∈J)dim L_i). Then S(D)=E(D), where S(D)=Zar_C[(Σ_i ∂_(L_i)D_i)∩∂D] uses the relative boundary in each summand span. Thus S(D) is irreducible under this explicit GP condition.
For every feasible prescribed full-rank summand type, GP is a nonempty real Zariski-open condition. Distinct Vandermonde columns witness every subset rank simultaneously. As in the earlier discotope work, the total span may be used as the ambient space.
Outside generic position, S(D) can strictly contain E(D). Two identical unit xy-discs plus one unit xz-disc give e_3∈S(D) while a polynomial vanishing on E(D) has value 16 at e_3. The separate equality theorem is therefore essential to the earlier S-conjecture application.
Why the result holds
Write D_i=A_i(B^(m_i)) with A_i injective. The normal u gives a unique summand support point precisely when A_i^T u≠0, with point A_i A_i^T u/||A_i^T u||. Exposed faces add, so the actual exposed-point set is exactly the image of their summed support map on the complement of the kernels.
Every excluded kernel has codimension at least two, so a two-segment detour makes the real normal domain connected. Positive real radicands make the support map real analytic without a global complex square-root branch.
If a product of two complex polynomials vanishes on the analytic image, one factor vanishes on a nonempty open part of the domain. Analytic uniqueness then forces that factor to vanish throughout the connected domain. The complex vanishing ideal is prime and the closure irreducible.
For S=E, a supporting normal kills a subset of summand spans. GP forces those spans to form a direct sum inside its perpendicular hyperplane. Their joint transpose is onto, allowing one normal perturbation to realize all prescribed unit relative-boundary parameters simultaneously. The resulting exposed points converge to the original boundary sum.
Scope and limitations
The general theorem assumes each summand has actual dimension at least two. A rank-one interval has two exposed points, and a planar disc plus a nonzero segment has two distinct conic components, so rank-one summands cannot be included indiscriminately.
Equality S=E is asserted under full dimensionality and the every-subset GP condition, with explicit ambient-span reduction. It is not asserted for arbitrary nongeneric sums; the repeated-disc example separates the two varieties algebraically.
The paper establishes no degree formula, birationality statement or irreducibility of the entire complex critical locus. General convex summands, infinite sums and other scope extensions are not part of its theorem.
The connected analytic-image principle, support-gradient parametrization and known discotope special cases are prior work. The contribution is their unrestricted exposed-point application and the generic identification bridge. Historical first priority and current worldwide open status are not asserted.
Status and AI use
Unrefereed preprint, version 1.0, developed with extensive generative-AI assistance. No human peer review, formal proof certification or first-priority claim is asserted.
Generative AI tools were used extensively in proof development, literature work, manuscript preparation, adversarial review and symbolic-check design. The named author is responsible for the submitted work. Internal AI checking is not human refereeing or formal proof certification; finite symbolic controls support examples and formulas rather than the universal proof.
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Cite this paper
Alec Kriebel. (2026). Irreducibility of exposed-point varieties of discotopes (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23074543
Background and prior work
- F. Gesmundo and C. Meroni, The Geometry of Discotopes, Le Matematiche 77(1) (2022), 143–171: Definition 3.3 and Conjecture 8.2 concern S. Theorems 4.3/6.1 establish generic low boundary-dimension-sum and rank-two cases; Remark 3.4 covers all full-dimensional summands. Section 5 already uses support gradients; Section 2 gives bases genericity and ambient-span reduction.
- W. Kucharz and K. Kurdyka, Rationality of semialgebraic functions, Analytic and Algebraic Geometry 2, Łódź University Press, 2017, 85–96: Proof of Proposition 4.3, p.95 explicitly records the real-Zariski connected analytic-image principle. The manuscript supplies the complex-coefficient prime-ideal argument.
- C. Meroni, Two convex conjectures for different flavours, New Directions in Real Algebraic Geometry, OWR 20 (2023), Report 15/2023, 829–832: Conjecture 1, p.830 defines the exposed-point closure E and asks its irreducibility for generic discotopes with all summands of dimension at least two. The first theorem removes genericity.