This is a joseki taken from http://eidogo.com/

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[go]$$B
$$ -------------------------
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . 5 3 4 . . . . . .
$$ | . . 1 9 2 . 6 . . 7 . .
$$ | . . 8 0 . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .[/go]

and then

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[go]$$B
$$ -------------------------
$$ | . . . . . . . . . . . .
$$ | . b a . . . . . . . . .
$$ | . . . X X O . . . . . .
$$ | . 5 X X O 2 O . . X . .
$$ | . 4 O O 1 . . . . . . .
$$ | . . . . 3 . 7 . . . . .
$$ | . . . . . . . . . . . .
$$ | . . 6 . . . . . . . . .
$$ | . . . . . . . . . . . .[/go]

According to the joseki in eidogo, the best move that white can make is to play at a, and black can respond correctly to live in the corner. However, it seems to me that if white plays at b instead, black cannot live in corner -- black needs to escape via the following sequence:

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[go]$$W
$$ -------------------------
$$ | . . . . . . . . . . . .
$$ | . 1 3 . 2 . 4 . . . . .
$$ | . . . X X O . . . . . .
$$ | . X X X O O O . . X . .
$$ | . O O O X . . . . . . .
$$ | . . . . X . X . . . . .
$$ | . . . . . . . . . . . .
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .[/go]

Would there be any way for black to live in the corner after W1?

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[go]$$W
$$ -------------------------
$$ | . 0 . a . . . . . . . .
$$ | 5 1 2 8 7 9 . . . . . .
$$ | 4 3 . X X O . . . . . .
$$ | 6 X X X O O O . . X . .
$$ | . O O O X . . . . . . .
$$ | . . . . X . X . . . . .
$$ | . . . . . . . . . . . .
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .[/go]

is the key point. After :wb10: it is at least a seki for black (white 'a' not sente).

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[go]$$W
$$ -------------------------
$$ | . 3 . . . . . . . . . .
$$ | 4 2 1 c b . d . . . . .
$$ | . . . X X O . . . . . .
$$ | . X X X O O O . . X . .
$$ | . O O O X . . . a . . .
$$ | . . . . X . X . . . . .
$$ | . . . . . . . . . . . .
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .[/go]

So initially it is important for white to run out first ('a' is a good move). We often see the two exchanges in the corner, making sure that white has some sente outside (for example white 'b', black 'c', then white 'd' is sente against the corner - this helps the eye space of the white group)

Thank you for your explanation!

I am thinking about the following alternative sequence, and whether this is playable for white or not:

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[go]$$W
$$ -------------------------
$$ | . 7 . 9 0 . . . . . . .
$$ | 5 1 2 . 8 . . . . . . .
$$ | 4 3 . X X O . . . . . .
$$ | 6 X X X O O O . . X . .
$$ | . O O O X . . . . . . .
$$ | . . . . X . X . . . . .
$$ | . . . . . . . . . . . .
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .[/go]

followed by

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[go]$$W
$$ -------------------------
$$ | 6 O 1 O X 5 . . . . . .
$$ | O O X 4 X 3 . . . . . .
$$ | X O 2 X X O . . . . . .
$$ | X X X X O O O . . X . .
$$ | . O O O X . . . 7 . . .
$$ | . . . . X . X . . . . .
$$ | . . . . . . . . . . . .
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .[/go]

One sequence that white would live on top is

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[go]$$B
$$ -------------------------
$$ | X . . . X O . . . . . .
$$ | . . X X X O . 8 6 7 . .
$$ | X . X X X O . . 4 5 . .
$$ | X X X X O O O 2 3 X . .
$$ | . O O O X . . 1 O . . .
$$ | . . . . X . X . . . . .
$$ | . . . . . . . . . . . .
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .[/go]

and this sequence also allows white to live on top:

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[go]$$B
$$ -------------------------
$$ | X . . . X O . . . . . .
$$ | . . X X X O . 8 . . . .
$$ | X . X X X O . 5 6 7 . .
$$ | X X X X O O O 1 4 X . .
$$ | . O O O X . 3 2 O . . .
$$ | . . . . X . X . . . . .
$$ | . . . . . . . . . . . .
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .[/go]

However, white gets a much smaller territory when compared with the joseki in eidogo.... so this is probably undesirable for white then.

Would black have a even strong response in this case?

Black would play better moves, namely

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[go]$$W
$$ -------------------------
$$ | . 7 . . . . . . . . . .
$$ | 5 1 2 . 0 8 9 . . . . .
$$ | 4 3 . X X O . a . . . .
$$ | 6 X X X O O O . . X . .
$$ | . O O O X . . . . . . .
$$ | . . . . X . X . . . . .
$$ | . . . . . . . . . . . .
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .[/go]

And black is alive and white didn't get the useful moves on the outside in sente, but just died inside black's territory. The peep at a will be annoying.