With increasing temperatures, it is likely that precipitation extremes increase as well. While, on larger spatial and longer temporal scales, the amplification of rainfall extremes often follows the Clausius-Clapeyron relation, it has been shown that local short-term convective precipitation extremes may well exceed the Clausius-Clapeyron rate of around 6.5%/K. Most studies on this topic have focused exclusively on the intensity aspect, while only few have examined (with contradictory results) how warmer and moister conditions modulate the spatial characteristics of convective precipitation extremes and how these connect to increased intensities. Here we study this relation by using a large eddy simulation model. We simulate one diurnal cycle of heavy convective precipitation activity based on a realistic observation-based strongly forced case setup. Systematically perturbed initial conditions of temperature and specific humidity enable an examination of the response of intensities and spatial characteristics of the precipitation field over an 8° dew point temperature range. We find that warmer and moister conditions result in an overall increase of both intensities and spatial extent of individual rain cells. Colder conditions favor the development of many but smaller rain cells. Under warmer conditions, we find a reduced number of individual cells, but their size significantly grows along with an increase of intensities over a large part of a rain cell. Combined, these factors lead to larger and more intense rain cells that can produce up to almost 20% more rain per degree warming and therefore have a large impact.