There is a tiny doorpost on the inside of the door but there's not enough room for a slanted placement. On the outside the door is flush with the wall so there is no doorpost to place it on at all, even vertically.
Similar to these doors:
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Can it be posted while standing upright? – chacham Nisan Dec 28 '18 at 6:36
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@chachamNisan on the outside the door is flush with the wall – larry909 Dec 28 '18 at 7:28
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It is important to ask these questions to a rabbi, and to send pictures if asking remotely, as small details can have important consequences.
In general, if there is no space on the doorpost, one can drill into it (and I saw new appartments in Israel where every mezuza was drilled into the doorposts - some prefer it this way halachically).
OU writes
Place the mezuza outside the door but within the door frame. If not possible, you may place the mezuza inside the door frame.
Note: You may recess a mezuza into the door frame.
TorahTots writes
In extenuating circumstances, such as the aforementioned examples, it would be preferred to drill a hole, less than a handbreadth deep, into the doorpost and place the mezuza into the groove.
If that is not possible, then it would be permitted to affix the mezuza behind the door provided it was placed on the doorpost under the lintel. [see picture on top of p. 4]
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I can't visualize these places in your answer: 1) "... If there is no space on the doorpost one can drill into it..." 2) OU writes: "it's not possible you may place the mezuzah inside the door frame you may recess a mezuzah into the door frame." 3) "it would be preferred to drill a hole...into the doorpost...into the groove." 4) "afix the mezuzah behind the door provided it was placed on the doorpost under the lintel." – larry909 Dec 28 '18 at 13:50
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1@larry909 the first three places are INTO the doorpost if there is space to drill a hole for the mezuza into its width or depth. If this is not possible see the drawing on top of page 4 of the TorahTots link (apologies I had originally forgotten the link but corrected it now). See here – mbloch Dec 28 '18 at 13:59
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@mbloch you mean on the outside, even if it's not a recessed doorpost as in the photo in my post? So the mizuzah will be facing outside. – larry909 Dec 28 '18 at 14:01
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@larry909 the second opinion from TorahTors is indeed on the outside, did you see their drawing (figure 3, top of p. 4)? – mbloch Dec 28 '18 at 14:02
