So, we agree that one ten minute confirmation represents x amount of work, yes? The way we'd have faster blocks (and thus confirmations) is to cut the amount of work required to find a block (on average). This is adjusted via the target difficulty (raise difficulty when average block time <10 minutes, lower difficulty when the average block takes more than 10 minutes to find).
So, when we make blocks happen faster, for example 10x faster (1 minute target blocks), difficulty is 1/10th the 10 minute difficulty (assuming it's a linear scale, don't recall offhand whether it is or not). This means that each block represents 1/10th the work to secure your money,
I agree with everything so far.
and thus you need 10 1 minute confirmation to represent 1 10 minute confirmation.
I disagree with this.
The likelihood of an attacker double spending a transaction with 10 one-minute confirmations is much lower than an attacker double spending a transaction with 1 ten-minute confirmation.
Security comes from the cost of executing a successful attack exceeding the payout from a successful attack. If the probability of an attack being successful is reduced, that means the cost of executing a successful attack increases, because it requires more attempts. Double spending a 10-one-minute-confirmation tx has the same payout for an attacker as double spending a 1-ten-minute-confirmation tx, but is harder (more costly) to execute.