If a rational preference relation over simple lotteries $succsim$ are convex then they satisfy independence?

1. It's well known that if $succsim$ satisfies independence, then it is also convex.

Since $succsim$ satisfies independence, $Lsuccsim L^{'} iff alpha L+(1-alpha)L^{''}succsim alpha L^{'}+(1-alpha)L^{''}$ for all $alpha in [0,1]$ and $ L, L^{'}, L^{''}in mathfrak{L} $

Convexity requires:

$Lsuccsim L^{''}$ and $L^{'}succsim L^{''} Longrightarrow alpha L$ + $(1-alpha)L^{'} succsim L^{''}$ for all $alpha in left[0,1right]$ and $ L, L^{'}, L^{''}in mathfrak{L} $

Pick any $ L, L^{'}, L^{''}in mathfrak{L} $ with $L succsim L^{''}$ and $L^{'} succsim L^{''}$. By completeness, $L succsim L^{'}$ or $L^{'} succsim L$ or both. Without loss of generality, assume $L succsim L^{'}$. Then, by independence, for all $alpha in [0,1]:$

$ alpha L$ + $( 1- alpha) L^{'} succsim alpha L^{'} + (1-alpha)L^{'} = L^{'} succsim L^{''} $, which we wanted to show.

2. Is it true if $succsim$ are convex then they satisfy independence?

On the other side, convexity does not imply independence. To see this, take a look at Figure (see below). Triangles are indifference curves and arrows show the direction in which utility increases. It is convex, but not independent.