Inner City in the sense of urban geography does not refer to the city centre but to a ring - the classical shape, though less common than it used to be - of deprived neighbourhoods around the city centre. It replaced (and is arguably a euphemism for) the older term 'slum'. There isn't really an English term for deprived suburb, but there is a French one and we might as well use it: banlieue.
And that's why I said near the urban core. Most American big city downtowns (as we call them, not sure if that term is used in the UK) are perfectly safe and, in addition to hosting the traditional officespace, are full of trendy, supercool types living in expensive lofts next to wine bars and small plate restaurants. Even in Memphis, the least supercool city in America. Outside of stray homeless people, of course poors don't live downtown. They often work there in restaurant kitchens and hotels though.
You have Graceland and Beale Street; you're far more supercool than, say, Fresno or Jacksonville.
As for the OP's question... Gustaf has the right of it.