(picture from [1], but the color added by the author.)
So we would like to know if an antenna in resonance gives the minimum VSWR, under the assumption that only the imaginary part of the feed point impedance varies when when we change the element length.
Let’s assume for the moment that the system impedance Z0 is 50 ohm, and if we further assume that the feed point impedance is given by "Z=50+j*X", then Z is always on the r=1 circle (green), and in this case obviously Z is most close to the origin (red dot) when X=0.
Actually in this particular case, Z becomes exactly 50 ohm when X=0, and coincides with Z0, thus the complex reflection coefficient gamma is 0, which means rho is also 0 and VSWR=1.