I think slide as trivial case of natatime might fit here:

use strict;
use warnings;
use feature 'say';
use List::Util qw/ min pairmap /;
use List::MoreUtils qw/ slide /;

say
    min
    slide { $b - $a }
    sort  { $a <=> $b }
    map   { $_, $_ + 60 * 24 }
    pairmap { 60 * $a + $b }
    map   /(\d\d):(\d\d)/,
    @$_
    for 
        [ qw/ 00:00 23:55 20:00 /],
        [ qw/ 01:01 00:50 00:57 /],
        [ qw/ 10:10 09:30 09:00 09:55 /]
;
[download]

(I'm so lazy I just duplicated all the clock readings to the next 24h, sorry. Second task could perhaps (?) be so more interesting for product instead of sum, if negative integers are allowed, but this is just a fleeting thought, I may be wrong)