Scalar Curvature Question [?65]: Is the regular Euclidean $3$-simplex mean-convexly extremal

Definitions and mathematical statement

Let $\Delta\subset{\mathbb{R}}^3$ be a regular Euclidean tetrahedron, with corresponding faces and edges fixed. In the interior-dihedral convention its common angle is $\theta_0=\arccos(1/3)$. The source asks whether $\Delta$ is mean-convexly extremal, by asking about a competing simplex $P$ mapped facewise to $\Delta$ without decreasing distances.

Under a face-metric reading, the distance requirement would be \[ d_{\Delta_i}(f_i(x),f_i(y))\geq d_{P_i}(x,y) \qquad(x,y\in P_i) \] for each corresponding face map $f_i:P_i\to\Delta_i$. This retains the stated nondecreasing direction; it is not replaced by distance contraction. The competitor is to have nonnegative face mean curvatures and dihedral angles no larger than the corresponding angles of $\Delta$, with at least one strictly smaller.

The area-extremality-style question is therefore whether such a strict-angle competitor can exist. Affirmative mean-convex extremality would rule it out; a yes answer to the existence question would refute that extremality. The source's word "equivalently" joins opposite yes/no formulations, not identical affirmative assertions.

The competitor category is not defined: an ordinary straight Euclidean simplex, a curvilinear Euclidean domain, and a Riemannian simplex with corners give different problems. The source supplies no interior scalar-curvature condition, and none is invented. The face distance metric, face-map regularity/bijectivity, mean-curvature normal/sign, and interior versus exterior dihedral convention must be specified. For curved edges, the meaning of one angle being strictly smaller also needs a pointwise or edgewise convention. The source does not demand that every non-strict competitor be isometric to $\Delta$; that stronger rigidity claim is not added.

Short English statement

Can a facewise distance-nondecreasing comparison to the regular tetrahedron have mean-convex faces and no larger dihedral angles, with one angle smaller? Mean-convex extremality would forbid such a competitor, subject to the missing geometric conventions.

Sources

• Local UnsolvedMath registry, permanent ID 6700061, exact source code catalogue entry; proposed by Misha Gromov, 2017. Registry status open, classification OPEN-TRIAGE. Source printed tag [?65], occurrence 64.
• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?65], PDF page 61, https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf. Supplied normalized pdf-text extraction accessed July 29, 2026; incomplete author-hosted PDF. No PDF was accessed here.
• The supplied summary calls this a concrete mean-curvature/dihedral question and reports no resolution located. It supplies no precise later reference or missing definition of mean-convex extremality.
• Provenance flags current status and statement-reuse terms NEEDS_REVIEW; the source marks the passage as a numbered open item.