Translation:I am looking for an apartment which is in the city center.
Okay, so I did some research on both "die Innenstadt" and "the inner city" to try and remove any personal bias I may have -- I'm a Brit who, although familiar with the term, wouldn't class it as a part of my active vocabulary.
innerer Teil des Stadtgebietes größerer Städte, durch den meist die Hauptgeschäftsstraßen führen; City, Zentrum
So, it looks rather convincingly to refer specifically to the area of a large city that is centrally located and through which the busiest streets run.
I think it's also worth noting for future reference that "Downtown" is listed as a synonym. It should become clear later on, why I believe that is worth noting.
So, onto "the inner city".
First of all, if you look at the "Inner city (disambiguation)" wiki page* you will find clearly written at the top of the page:
The inner city is the central area of a major city.
Which aligns rather nicely to the definition from Duden, one has to say.
However, if you follow that link, it leads you to an article which begins as follows:
In the United Kingdom the term "inner city" has been used as a euphemism for lower-income residential districts in the city center and nearby areas. Sociologists sometimes turn this euphemism into a formal designation, applying the term "inner city" to such residential areas, rather than to geographically more central commercial districts.
Which seems to imply a socio-economic aspect to "inner city", which doesn't appear to be present with "Innenstadt".
To bring in a perspective from across the pond I thought I would go to the universally accepted and respected Urban Dictionary**; which had this to say on the subject:
A term used in the United States referring to a place in a large city that is mostly populated by poor minorities. It has nothing to do with the actual inner part of a city, and generally refers to a wedge-shaped area within the city limits that separates the downtown area from the suburbs...
The inner part of a city is the downtown area, but the part that is called "inner city" is the outside part.
So, it seems that "the inner city" tends to have a societal implication that "die Innenstadt" doesn't, and that, somehow (and almost certainly more thanks to English being weird rather than German), "downtown" is often a better translation for "Innenstadt" than "inner city".
*Yes, I know Wikipedia isn't the most reliable of sources, because anyone can write on any wiki page and say what they want; but there are moderators who clean up the mess at quite a rate, so with a pinch of salt I think we can look at what's there pretty objectively.
**What? You mean, some of the stuff on Urban Dictionary isn't supposed to be taken seriously?! Rubbish! But in all seriousness, though it is of course colloquial slang that makes up the vast majority of pages on UD, it's still useful, in my mind, to get an idea of how a term is interpreted and currently used (even if in some cases that use could be considered to be "wrong"). I guess I should also add a disclaimer here that on the Urban Dictionary things can get dicy pretty quickly if you're not careful (also the reason why I left out the links from the UD "inner city" page). So, there's my disclaimer.
That makes sense. I looked it up and saw the same definition among many different dictionaries implying areas in a city which have "social problems".
Didn't know about that though, I am not a native english speaker but find it quite usual for inner-city to only mean "central area of the city".