Convexity is what makes a player feel freedom of choice, while being steered down a predetermined path. For example, take any game in the mass effect series. You usually get a lot of different choices in conversations, even allowing you to choose if someone should live or die. If the person you killed in the conversation was supposed to do something meaningful in the story, he's usually replaced by somebody who will then do that meaningful thing.
You get choices on how to complete a mission, but the story will not continue until you've completed it. The pinnacle of this is actually the ending of mass effect 3. The game allows you to choose a lot of different things and in the end [SPOILER ALERT] you really only change the color of the explosion.
Convexity is a good thing, because it allows the player to have freedom of choice, while still following an awesome story.
Convexity is also a bad thing, because developers usually boast about you being able to choose your own destiny while really, your choices boil down to the same scenario. (Sometimes conversations let you choose to piss someone off or kiss his ass. Pissing him off will start a fight, while kissing his ass makes him angry... starting a fight)
Max Payne is an example of a game without convexity. In the first game, you will have the choice to kill an innocent person. Do so, and it's game over. That's no choice, that's just linear gameplay.
Max Payne 2 allows you to kill someone you believe is innocent. He's not. If you let him live, you'll have to kill him about 60 seconds later.
There are games where you get true freedom of choice (or something very close to it at least), but because the game has to be able to adapt to an infinity of scenarios, the gameplay usually isn't very deep (take spore for example; millions of planets to explore and they're all empty, boring pieces of land with little diversity.)
So really, it's not a good or bad thing. It depends on what you want your game to be. It is a huge impact on the flow of the game, so it's a good thing to consider when designing your game. If you're forcing a player down a predetermined path, it better be one hell of a ride.