Narrow Boards

Fig 2. Mirroring a triangle (or L shape) is a bad idea  

2 x Infinity board

There are three different variations for 2 x Infinity boards: cornerless, 2-corner, and 4-corner, as illustrated below:

2 x Infinity cornerless  

The minimal number of stones for a live group without corners is 6.

2 x Infinity with 2 corners  

The minimal number of stones for a live group with corners is 4. The 4-corner case is equivalent two of 2-corner boards, which can be considered independent since they are separated by an infinite distance.

2 x Infinity with corners

Black's opening options can be analyzed as follows. 1-1 leads to capture.

B 1-2, W 2-4, B 1-5 leads to stalemate  

If black starts with 1-2, 2-4 from white threatens to take the corner. Black 1-5 leads to stalemate. White must respond with 2-5 to avoid being confined to the 4th column (and being left with a bent four). From this point forwards, both players are forced to continue extending their lines indefinitely. Black must continue to avoid being cut off; white unable to hane due to the threat of black forming an L-shape (see Fig. 2.)

B 1-2, W 2-4, B 2-3 (First resistance, part 1)  

If black responds by 2-3, white can reply with 1-5. Black plays 2-6 to threaten white's empty hane. If white starts a fight (First resistance), black wins the capturing race.

B 1-2, W 2-4, B 2-3 (Second resistance)  

But if white completes the 4th column (second resistance), black's mirrored triangle forces white to block black's miai. But now black can capture white.

B 1-2, W 2-4, B 2-3 (Third resistance)  

The same occurs if white takes the 5th column (third resistance).

3x Infinity

Minimal live group without corners  

The minimal number of stones for live group without corners is 7.

Minimal live group with corners  

With corners, the number is 4.