Does the L-sit:
overlap with the plank:
or do these two exercises complement each other well? Is there any benefit to holding planks that can't be obtained by holding L-sits?
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Both exercises have some overlap in terms of the muscles used, particularly internal abdominal muscles (if you maintain good posture in the plank), but I would say that they are significantly different exercises and both worth practising in their own right. In fact, beyond being static positions and involving abdominal contraction they are rather dissimilar.
I would consider the L-sit to be a harder exercise than the plank. Certainly most people can hold a plank for much longer (with a little practice) than an L-sit. For example, I have held a plank for just over 8 minutes, but would struggle to hold an proper L-sit for more than 15-20 seconds. That said maintaining a really good hollow bodied plank, with tight glutes and lower abdominals is challenging.
Here are some key differences between the two exercises:
You might want to work progressions of the plank such as: http://gymnasticswod.com/content/v-out (very hard, I find). The L-sit should help with leg position in somersaults, and the plank or rather a self-supporting hollow body position is very fundamental throughout gymnastics.