Convexity and the partition of the unity

Question

Let $C$ be a convex subset of the Banach space $X$ of the continuous functions $f:[0,1]\longrightarrow \mathbb{R}$, and consider $\lambda_{1},\ldots,\lambda_{n}:[0,1]\longrightarrow [0,1]$ a partition of the unity (i.e, each $\lambda _{i}$ is continuous and $\sum\lambda_{i}(t)\equiv 1$.

Then, fixed $f_{1},\ldots,f_{n}\in C$, Can we ensure that the function $y(t):=\sum_{i=1}^{n}\lambda_{i}(t)f_{i}(t)$ belongs to $C$?

For some convex sets, the answer is yes. Indeed, we can think in the set

$C:=\{f\in X: 0=f(0)\leq f(x)\leq f(1)=1\}$

But, is true this fact in general?

Many thanks in advance for your comments.

Answer 1

No, consider $C$ to be the set of all constant functions (which is clearly convex) and

$$\lambda_1(t)=t$$ $$\lambda_2(t)=1-t$$ $$f_1(t)=1$$ $$f_2(t)=0$$

Then $$y(t)=t$$ is not constant so it does not belong to $C$.