**Authors:**

(1) Edward Crane, School of Mathematics, University of Bristol, BS8 1TH, UK;

(2) Stanislav Volkov, Centre for Mathematical Sciences, Lund University, Box 118 SE-22100, Lund, Sweden.

## Table of Links

Reduction to the case of uniform geometry

All original points are eventually removed, a. s.

Acknowledgements and References

## Acknowledgment

The research of S.V. is partially supported by the Swedish Science Foundation grant VR 2019- 04173 and the Crafoord Foundation grant no. 20190667. S.V. would like to acknowledge the hospitality of the University of Bristol during his visits to Bristol. The research of E.C. is supported by the Heilbronn Institute for Mathematical Research. We would like to thank John Mackay for pointing us to the notion of uniform geometry in [7] and Rami Atar for pointing out the relevant Brownian bees model. Finally, we would like to thank the anonymous referee for many useful comments and recommendations, in particular the suggestion to explain what can be done in the non-convex case.

## References

[1] Addario-Berry, Louigi; Lin, Jessica; Tendron, Thomas. Barycentric Brownian bees, Ann. Appl. Probab. 32 (2022), no. 4, 2504–2539.

[2] Grinfeld, Michael; Volkov, Stanislav; Wade, Andrew R. Convergence in a multidimensional randomized Keynesian beauty contest, Adv. in Appl. Probab. 47 (2015), no. 1, 57–82.

[3] Durrett, Rick. Probability: theory and examples. Fifth edition. Cambridge University Press (2019).

[4] Kennerberg, Philip; Volkov, Stanislav. Jante’s law process, Adv. in Appl. Probab. 50 (2018), no. 2, 414–439.

[5] Kennerberg, Philip; Volkov, Stanislav. Convergence in the p-contest, Journal of Statistical Physics 178 (2020), 1096–1125.

6] Kennerberg, Philip; Volkov, Stanislav. A Local Barycentric Version of the Bak-Sneppen Model, Journal of Statistical Physics 182:42 (2021), 17 pp.

[7] Leonardi, Gian Paolo; Ritor´e, Manuel; Vernadakis, Efstratios. Isoperimetric Inequalities in Unbounded Convex Bodies. Memoirs of the American Math. Soc., no. 1354, Vol. 276 (2022).

[8] Siboni, Maor; Sasorov, Pavel and Meerson, Baruch. Fluctuations of a swarm of Brownian bees, Phys. Rev. E 104 (2021), 7 pp.

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