L2 Bonnet–Myers and dimension

Definitions and mathematical statement

Let $(X,d)$ carry Markov evolution laws $m_x^{*t}$ starting at $x$, at the admissible times $t$ of the evolution. On the usual interpretation, $T_1$ denotes the Wasserstein transport distance $W_1$ for $d$. The displayed strengthened estimate is \[ T_1(m_x^{*t},m_{x'}^{*t'})\le e^{-\kappa\min(t,t')}d(x,x')+ C\frac{(\sqrt t-\sqrt{t'})^2}{2d(x,x')}, \] for all distinct $x,x'$ and admissible nonnegative times $t,t'$, with $\kappa>0$ and finite $C>0$. Distinctness is required because of the denominator. The source associates to this hypothesis the diameter bound \[ \operatorname{diam}(X)=\sup_{x,x'\in X}d(x,x') \le\pi\sqrt{\frac{C}{2\kappa}}. \] The question concerns the meaning of $C$, rather than a new proof of that bound: does it reflect an intrinsic dimension, especially the parameter $n$ in Bakry--Émery $CD(K,n)$?

For comparison, in a setting with a suitable Markov generator $L$, define \[ \Gamma(f,g)=\tfrac12\left(L(fg)-fLg-gLf\right),\qquad \Gamma(f)=\Gamma(f,f),\qquad \Gamma_2(f)=\tfrac12L\Gamma(f)-\Gamma(f,Lf). \] The finite-dimensional curvature-dimension inequality is \[ \Gamma_2(f)\ge K\Gamma(f)+\frac{(Lf)^2}{n} \] pointwise for all admissible test functions, with a specified finite dimension parameter $n>0$. Comparing these parameters requires the same generator/time normalization and a precise class of spaces; neither $K=\kappa$ nor $C=n$ is asserted by the source.

Moreover, any larger $C$ also satisfies the transport estimate when one $C$ does. An intrinsic interpretation would therefore have to specify an optimal coefficient, or another normalization selecting $C$. The source asks for a relation without defining this selection or a proposed formula. The supplied research summary calls the diameter bound established and the specific dimension identification unresolved, while distinguishing known Bakry--Émery and discrete comparison results.

Short English statement

Does the coefficient $C$ in the stated two-time transport bound represent a dimension, and how does it relate to the $n$ in $CD(K,n)$? The transport and generator normalizations and the choice of coefficient must be fixed.

Sources

• Local UnsolvedMath registry, permanent ID $4000018$, exact source code catalogue entry, "L2 Bonnet--Myers and dimension."
• Yann Ollivier, Discrete Ricci curvature: Open problems (2008), Problem R; https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf. Recorded access 2026-07-29; pdf-text extraction.
• Supplied research summary attributes the displayed Bonnet--Myers-type bound to Ollivier's article, analogous Bakry--Émery bounds to Bakry--Qian, and later discrete Laplacian-comparison bounds to Münch. Complete references and exact normalization conventions are not supplied.