Discrete scalar curvature

Definitions and mathematical statement

Let $(X,d)$ be a metric space with a probability transition law $m_x$ at each point. For $x\ne y$, Ollivier coarse Ricci curvature is \[ \kappa(x,y)=1-\frac{W_1(m_x,m_y)}{d(x,y)}, \] where $W_1$ is the minimum expected distance over probability couplings, assuming finite first moments. The proposed scalar-curvature candidate is \[ S(x)=\int_X\kappa(x,y)\,\mathrm{d}m_x(y), \] possibly replaced by a distance-weighted average \[ S_w(x)=\int_X w(d(x,y))\kappa(x,y)\,\mathrm{d}m_x(y). \] These formulas require integrability. The unweighted expression also requires a convention at $y=x$ if $m_x$ has an atom there, because the displayed coarse-curvature formula is undefined on the diagonal. The source gives no diagonal convention or particular weight $w$.

The problem asks whether this averaged quantity has useful geometric or probabilistic properties, with volume-growth control offered as an example. A volume-growth question would concern the mass of metric balls $B(x,r)=\{y:d(x,y)\leq r\}$ with respect to a specified global measure, such as counting measure on a locally finite graph. The source supplies neither that global measure nor hypotheses and a quantitative volume comparison. Accordingly this is a proposal to develop and test a scalar-curvature notion, not a stated universal volume bound.

Short English statement

Study the average of coarse Ricci curvature over a point's transition law, possibly weighted by distance. Does this proposed scalar curvature control useful geometry or probability, for example growth of metric balls?

Sources

• Local UnsolvedMath registry, permanent ID 4000017, exact source code catalogue entry, "Discrete scalar curvature"; proposed by Yann Ollivier, 2008. Registry status open, classification OPEN-TRIAGE.
• Yann Ollivier, Discrete Ricci curvature: Open problems (2008), Problem Q, https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf. Supplied pdf-text extraction accessed July 29, 2026; the item is presented as open, and no later authoritative resolution check is recorded. No PDF was accessed here.
• The supplied research summary describes the functional as a proposal and reports no verified established theory or volume-growth control for this particular quantity. This is a limited supplied search assessment, not a claim that no such result exists.
• Provenance flags status and rights NEEDS_REVIEW: public access was verified, but redistribution permission was not established.