  Context. It is unclear why there is a rather sharp boundary in sunspots
between the umbra and the penumbra. Both regions exhibit magnetoconvection,
manifesting 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 size and brightness of
convective features in the penumbra.
