The real point of "flat-fingered" playing has been overlooked. Only Marik touched upon it.
Flat-fingered is an over-simplification. No one, not even Horowitz, plays "flat-fingered," literally and strictly speaking.
A teaching assistant of Menachem Pressler coached me some years ago on the Schumann Concerto. And she stressed the concept of "scratching" the keys. An approach that is compatible with any hand position -- arched or flat.
To illustrate: it is obvious that in Bach or Mozart, where wide hand extensions are not the rule, a so-called "arched hand" position is the more natural postion. But, even in this position, the attack on the keys is initiated by a scratching motion, i.e. a finger stroke that emanates from the tip of the finger. Following this initial stroke, the finger then pulls the energy of this finger-tip stroke into the palm. It feels, initially, like a staccato stroke, but it is adjusted to the requirement of the pitch. For legato, the stoke is prolonged, drawn-out, but it still begins its life as a "scratch."
Great pianists like Horowitz (and I can't believe there are people on this forum with the arrogance and ignorance to disparage him) may appear to employ "flat-fingered" technique, but, in reality, the scratching approach is part of the initial key stroke. In other words, there is the inherent feel to the key stroke of an "arched" hand. But the "arched hand" is manifested in the flattened palm with the energy channeled into the fingertip in the "scratching" key stroke.
I have an average size hand. But in working on Rachmaninov (currently the Second Concerto), I encounter very wide extensions that require me to play with a "flat-fingered" approach. But, still, the primary energy of my hand is concentrated in the fingertip.
Pianists, like "amelia" who adhere to a dogma professed by their teachers, only exacerbate this problem of arched hand over flat-fingered hand. Flat-fingered playing is not only unavoidable but natural and expectable, especially if one has a small hand. Observe YOUR OWN hand. Feel the energy needed to coax beautiful sound from the instrument. It is your fingertip that is the point of contact. Your body is behind that fingertip. That is the point. Not arched or flat. It doesn't matter. Concentrate on the FINGER-TIP.