Dice Question Streamline Icon: https://streamlinehq.com

Guaranteeing the one-dimensional cumulative bound along needles in higher dimensions

Identify conditions in higher-dimensional settings under which, for almost every needle in the Cavalletti–Mondino localization of an RCD(0,N) space (N>1), the one-dimensional cumulative distribution function R of the pushforward measure along the Busemann function satisfies the bound R(x) ≤ R(0)·(1 + (c/N)·x)^N for all x in the needle, where c = w(0)/R(0) and w is the density of the pushforward measure.

Information Square Streamline Icon: https://streamlinehq.com

Background

A key step in the one-dimensional proof of Grünbaum’s inequality is the concavity of R{1/N}, which yields the explicit cumulative bound R(x) ≤ R(0)(1 + (c/N)x)N. This bound drives stability and rigidity arguments.

Extending this bound from the one-dimensional setting to almost every needle in higher-dimensional localization would strengthen the multi-dimensional analysis, but the authors state they do not know a suitable condition that guarantees this property.

References

In higher dimensions, however, we do not know a suitable condition guaranteeing eq:R on almost every needle.

eq:R:

R(x)≤R(0)(1+cNx)N,c:=R′(0)R(0)=w(0)R(0)>0.R(x) \le R(0) \biggl( 1+\frac{c}{N}x \biggr)^N, \qquad c:=\frac{R'(0)}{R(0)} =\frac{w(0)}{R(0)}>0.

— A generalization of Grünbaum's inequality in RCD$(0,N)$-spaces (2408.15030 - Brunel et al., 27 Aug 2024) in Remark 3.5(a), Section 3.2 (One-dimensional analysis)