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| openaire: EC/H2020/633053/EU//UFOS Funding Information: Acknowledgements. This work benefited from the Hinode sunspot database at MPS, created by Gautam Narayan. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 695075) and has been supported by the BK21 plus program through the National Research Foundation (NRF) funded by the Ministry of Education of Korea. Hinode is a Japanese mission developed and launched by ISAS/JAXA, collaborating with NAOJ as a domestic partner, NASA and STFC (UK) as international partners. Scientific operation of the Hinode mission is conducted by the Hinode science team organized at ISAS/JAXA. This team mainly consists of scientists from institutes in the partner countries. Support for the post-launch operation is provided by JAXA and NAOJ (Japan), STFC (UK), NASA, ESA, and NSC (Norway). Publisher Copyright: © B. Löptien et al. 2021. ; Context. It is unclear why there is a rather sharp boundary in sunspots between the umbra and the penumbra. Both regions exhibit magnetoconvection, which manifests in penumbral filaments in the penumbra and in umbral dots in the umbra. Aims. Here we compare the physical properties of umbral dots and penumbral filaments. Our goal is to understand how the properties of these convective features change across the boundary between the umbra and the penumbra and how this is related to the rapid increase in brightness at the umbra-penumbra boundary. Methods. We derived ensemble averages of the physical properties of different types of convective features based on observations of two sunspots with Hinode. Results. There are strong similarities between the convective features in the outer parts of the umbra and the ones in the penumbra, with most physical parameters being smooth and continuous functions of the length of the features. Conclusions. Our results indicate that the transition in brightness from the umbra to the penumbra is solely caused by an increased effectiveness of magnetoconvection within individual convective cells. There is no significant difference in the number density of convective elements between the outer umbra and the inner penumbra. Penumbral filaments exhibit a larger area and a higher brightness compared to umbral dots. It is still unclear how exactly the underlying magnetic field causes the increase in the size and brightness of convective features in the penumbra. ; Peer reviewed