Set of consumption over all period is convex in $mathbb{R}^T_+$?

Assuming the following:

• $ S > 0 $, $T > 0 $,
• $ c(S) = left{cinBbb{R}^T_{+} : sum_{t=1}^T{c_t} = S right} $, and
• $ U(cdot) $ continuous.

Let's start with convexity. To demonstrate convexity, we need to show that for any two points $c^1$ and $c^2$ in $c(S)$, any linear combination of the two is also an element of $c(S)$.

Let $lambda in [0,1]$, $c^1 in c(S)$, $c^2 in c(S)$. Then, we have

$$ sum_{t=1}^T{lambda c^1_t + (1-lambda) c^2_t} $$ $$ = sum_{t=1}^T{lambda c^1_t} + sum_{t=1}^T{(1-lambda) c^2_t} $$ $$ = lambda sum_{t=1}^T{c^1_t} + (1-lambda) sum_{t=1}^T{c^2_t} $$ $$ = lambda S + (1-lambda)S $$ $$ = S $$ $$ Rightarrow lambda c^1 + (1-lambda) c^2 in c(S) $$

Since any linear combination of any two points in c(S) is itself in c(S), we have c(S) convex.

For non-emptiness, we just need to show that a single point exists in c(S): $$ sum_{t=1}^T{frac{S}{T}} = S $$ $$ Rightarrow left{ frac{S}{T} right}^T in c(S) $$

If $U(cdot)$ is continuous on $c(S)$, then continuity of $W(c)$ follows from the triangle inequality (see e.g., here or here). If we don't know more about $U(cdot)$, then there's not much we can say about $W(c)$. Berge's Theorem of the Maximum has to do with the continuity of an optimized function, not the sum of functions.