Irreducibility of Exposed-Point Varieties of Generic Discotopes

Definitions and mathematical statement

Let $d\geq 2$ and $N\geq 1$ be integers. A disc in ${\mathbb{R}}^d$ means a linear image of a Euclidean closed unit ball: for each $i$, choose an integer $m_i\geq 2$ and a linear map $A_i:{\mathbb{R}}^{m_i}\to{\mathbb{R}}^d$, and put \[ B^{m_i}=\{u\in{\mathbb{R}}^{m_i}:\|u\|_2\leq 1\},\qquad D_i=A_i(B^{m_i}),\qquad r_i=\dim D_i=\operatorname{rank}A_i. \] Assume $r_i\geq 2$ for every $1\leq i\leq N$. The associated discotope is the Minkowski sum \[ D=D_1+\cdots+D_N=\{x_1+\cdots+x_N:x_i\in D_i\text{ for every }i\}. \] Its set of exposed points is \[ \operatorname{Exp}(D)=\left\{x\in D:\text{there exists }v\in{\mathbb{R}}^d\setminus\{0\}\text{ such that }\langle v,x\rangle>\langle v,y\rangle\text{ for every }y\in D\setminus\{x\}\right\}. \] To display the unresolved base-field convention explicitly, for $K\in\{{\mathbb{R}},{\mathbb{C}}\}$ define \[ I_K=\{f\in K[x_1,\ldots,x_d]:f(x)=0\text{ for all }x\in\operatorname{Exp}(D)\},\qquad E_K=V_K(I_K)\subseteq K^d. \] Here the real points are embedded in ${\mathbb{C}}^d$ when $K={\mathbb{C}}$. A nonempty Zariski-closed set is irreducible if it is not the union of two proper Zariski-closed subsets.

Supplied conjecture, with unresolved conventions. For every generic discotope $D$ satisfying $\dim D_i\geq 2$ for all $i$, the Zariski closure $E$ of $\operatorname{Exp}(D)$ is irreducible.

The supplied record does not define the parameter space or exceptional set underlying "generic," and it does not explicitly specify whether $E$ means $E_{{\mathbb{R}}}$ or $E_{{\mathbb{C}}}$. Its background discussion uses ${\mathbb{R}}^d\subset{\mathbb{C}}^d$, suggesting a complex algebraic interpretation, but this does not settle the convention. Thus the conjecture above preserves the supplied target; it does not assert the conclusion for both fields or replace genericity by a newly chosen hypothesis. No additional full-dimensionality assumption is imposed.

Short English statement

A discotope is a Minkowski sum of linear images of Euclidean unit balls. The supplied conjecture says that, for a generic choice of discs each having dimension at least two, the algebraic closure of the exposed points is irreducible. The meaning of genericity and the field of Zariski closure require clarification before this becomes a fully specified statement.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30005473$, source code catalogue entry; title: Irreducibility of Exposed-Point Varieties of Generic Discotopes. The record gives proposed year 2023 and names no proposer.
• New Directions in Real Algebraic Geometry (2023), Oberwolfach Reports, pp. 813--881, DOI 10.4171/owr/2023/15. Supplied report and literature-source link: https://doi.org/10.4171/owr/2023/15. Oberwolfach Reports document mathematical workshops and their open-problem discussions.
• Gesmundo--Meroni, The Geometry of Discotopes. Supplied literature-source link: https://arxiv.org/abs/2111.01241.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and reports irreducibility results for two-dimensional discs and several low-codimension cases, while describing the conjecture for generic discotopes with higher-dimensional discs as open. This is supplied dated curation, not independently verified; time-sensitive status and the cited claims require independent verification.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance claims, not verification performed for this draft.