Hans: If white starts she can win.
1. White first:
MikeTerry: Capturing the enemy stones in your eye space earlier than you need to is often a mistake. Here, capturing with allows Black to start filling liberties again one move earlier than if were at in the next diagram.
White wins by one move.
MikeTerry: If in the first diagram above had taken a shared liberty rather than capturing the corner stones, White would have won the semeai by two moves rather than one. However, I believe you have missed the optimum line in the sequence above, and the situation is more complicated than this. (I think with White first, the status strictly depends on the ruleset in use!)
Suggestion: concentrating on the Black first situation will help clarify the White first situation...
I should add here that this is a position I have made up and analysed myself, and I've been known to produce total gibberish analyses in the past! So I apologise to everyone in advance if I'm doing it again :)
MikeTerry: at a?
The extra liberty at a allows White to play at and live. Alternatively, if at (the normal move to try and kill this eye shape in the corner), White can play at and make a big eye to win the semeai.
Bill: Oops! ;-)
So Black must fill all the shared liberties before completing the rectangular-six shape, as in Unkx80's latest attempt somewhere below.
This is a ko.
More analysis later...
unkx80: More analysis, as promised. To answer the at a question.
When do we fight this ko?
This is an approach ko.
unkx80: Try again...
Back to the throw-in ko.
MikeTerry: Right! This is the best sequence I've found, and leads to a direct ko. White will proceed to fill the Black eye liberties on the right, forcing Black to start the ko, and so Black will have to make the first ko-threat.
So unless anyone has a better solution, going back to the original problem I believe if Black starts, the outcome is a direct ko.
That leaves us with the White-first situation to sort out, which follows similar ideas to those above, but has it's own peculiar feature to be discovered! :)