Another boat comes into the mark from a totally different angle — can I shut the door?

No. The instant the first of you reaches the zone — three hull lengths from the mark — and you're overlapped, Rule 18.2(a) makes the outside boat give the inside boat mark-room. Being the leeward or right-of-way boat under Rule 11 does not save you — the mark-room rules take over inside the zone. Hold your ground, touch, and you're the one disqualified. (World Sailing Case 12.)

<iframe width="560" height="315" src="https://www.youtube.com/embed/jfhQaiLd1g4" title="Sailing Kiwi" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
<p><a href="https://sailing.kiwi.nz/windward-mark/case-12-you-cant-come-in-here/">Source: Sailing Kiwi</a></p>

Next on this shelf — The Windward Mark