Would half a wavelenght + a few cm be better if the antenna gets proper impedance matching? How about some integer multiple of half the wavelength?
There are two key factors to consider:
1) Antenna radiation efficiency - it is generally proportional to the antenna's physical dimensions. Since radiated power distributed across space, a larger antenna can collect and radiate more energy per oscillation cycle. However, this represents an upper limit, actual efficiency depends on design and losses.
2) Antenna losses - these include resistive losses and losses in the impedance matching network. A half-wavelength dipole is naturally resonant, meaning its input impedance is mostly resistive, minimizing the need for reactive matching components. Every impedance matching circuit introduces losses, and a higher impedance transformation ratio generally results in greater power dissipation.
Regarding your question, a length of half a wavelength plus a few centimeters may not provide significant advantages unless it improves impedance matching. Integer multiples of half a wavelength maintain the self-matching property but may not always be practical due to other design constraints.
It is also worth mentioning that if you ignore losses in the matching network and the antenna itself, you can assume that the antenna's efficiency is directly proportional to its size. However, there is a lower size limit due to capacitive/inductive coupling effects in the reactive near field region, even if you manage to match the source to the antenna without losses. This limit can be roughly estimated as λ / (2*π). If you reduce the antenna size below this limit, you will still have to keep the space in the area of this size empty, otherwise any objects that get into the antenna area at a closer distance will lead to strong losses, distortions of the radiation pattern and impedance of the antenna.
Thus, λ / (2*π) can be considered as the minimum size of the antenna, even if the physical design of the antenna will take up less space, its real working space still needs at least λ / (2*π) size.