  The Localized Waves are nondiffracting ("soliton-like") and
self-reconstructing solutions to the wave equations, and are known to exist
with subluminal, luminal and superluminal peak-velocities. The most studied
ones are the "X-shaped" superluminal waves; which are associated with a cone,
so that some authors [e.g., Walker and Kuperman, PRL 99 (Dec.2007) 244802] have
been tempted to link them with Cherenkov radiation. However, the "X-waves"
belong to a different realm, and exist even in the vacuum, independently of any
media, as verified in a number of papers [listed e.g. in "Localized Waves"
(J.Wiley; Jan.2008)]. We want to clarify the whole question on the basis of
rigorous formalism and clear physical considerations. In particular, by
explicit calculations based on Maxwell equations only, we show that: (i) the
X-waves exist also inside both the front and the rear part of their double cone
(that has nothing to do with Cherenkov's); (ii) they are to be found not via ad
hoc assumptions, but by use of strict mathematical (or experimental)
procedures; (iii) the ideal X-waves possess infinite energy, but finite-energy
X-waves can be easily constructed (even without space-time truncations): And we
do construct exact finite-energy solutions (totally free from
backward-traveling waves); (iv) an actual attempt at comparing Cherenkov
radiation with X-waves would lead one to consider the very different situation
of the (X-shaped, too) field generated by a superluminal point-charge, a
question exploited in previous papers [Recami et al., PRE 69 (2004) 027602]: We
show explicitly, here, that the point-charge would not lose energy in the
vacuum, and its field would not need to be continuously feeded by incoming
side-waves (as it is the case, indeed, for an ideal, ordinary X-wave).
