You seem to have some very conflicting objectives and I can't let this question stand as is when.
You say you don't want to change the shape and you don't want round corners but you don't say why.
The above answer will achieve what you are trying to do but it will not be a good fit to your underlying problem.
You say your points aren't very accurate. Do they get more inaccurate the further they are away from Germany? And do they do so in a linear fashion? and in a radially outward fixed direction? I cannot image that this would ever be the case and as it is the sole reason why you would want to enlarge your shape with an affine expansion, it's safe to say you should not be performing an affine expansion to fit your points.
E.g have a look at the pink and green polygon of the accepted answer. At one section their boundaries are touching. An affine expansion provides no advantage in being able to include inaccurate points here. Imagine what would happen to Sylt by an affine expansion? It would move and although it may catch some inaccurate points off the coast away from Germany, it would exclude points on the mainland side of the island if not potentially exclude the whole thing.
If; however, your inaccurate points are in error by an amount that is uncorrelated with their location, then what you DO want is buffering. Yes it will give round edges and yes it will change the shape but if you want to check if points are near enough then that IS the operation you want.