I asked about this before joining my current employer: I'd like to write a tetris clone and sell it on the AppStore (that was the specific example I used). I promise I won't use company resources. They said please don't. It's a grey area. Worst case you'll have to explain to a judge why your code doesn't infringe on anything the company owns or does. Even if it's something as trivial as a tetris clone.
You are however fairly safe if it's completely different from your work, for example, pottery or writing fiction (assuming your company does neither).
As for "as part of this employment relationship", I'm not a lawyer but I wouldn't assume that it applies only to what you do at the office. It might very well apply to anything while you are employed there. You need to clear the scope of that with HR or a lawyer.
You definitely don't want to base your company on shaky legal grounds. You might lose your company, your product, and your job. It probably won't look good in your Arbeitszeugnis either.
You are however fairly safe if it's completely different from your work, for example, pottery or writing fiction (assuming your company does neither).
As for "as part of this employment relationship", I'm not a lawyer but I wouldn't assume that it applies only to what you do at the office. It might very well apply to anything while you are employed there. You need to clear the scope of that with HR or a lawyer.
You definitely don't want to base your company on shaky legal grounds. You might lose your company, your product, and your job. It probably won't look good in your Arbeitszeugnis either.