I have a bunch of videos taken with my phone, and they're pretty massive given their length (eg. a 3 min video at 1920×1080 is over 400MB).

I presume this is because the phone favours battery life and quality over storage space, so I figure I could run them through Avidemux to reencode them and reduce their size while not compromising their quality too much. I realise that some degradation is inevitable. But since there's so much room for compression, I think it should be possible to avoid a great deal of perceptible degradation.

The videos are encoded in H264/AAC in an MP4 container. I can just copy the audio directly in Avidemux. But how do I deduce some approximate values for the H264 encoder to maintain the quality of the video? I don't think the H264 format keeps information about its input parameters. It's possible to choose a quality level that totally bloats the file size with no gain, and it's also possible to choose a quality level that makes the video terrible; is there some way to analyse the input to figure out a "sweet spot" where video size is reduced without affecting quality too much?

I realise that there's some degree of subjectivity in this question; what would I consider to be "perceptible" degradation? However, given that video encoders are traditionally parameterised by perceptual qualities, is it possible to come up with an approximate range for some basic parameters?