Well as the title says I would like to know if given a convex space $ X $, a $0$-cycle (equivalently any $0$-chain, right?), such that its augmentation is null is a boundary? All this in singular homology. I think a cone construction would do the trick but I do not know how. May be the argument is simpler using the reduced homology.
Thanks for any help you could provide.
Path-connectedness of $X$ is sufficient here. You can choose any point $x_0$ in $X$, and for each point $x_i$ in your $0$-chain $c=\sum\eta_i x_i$ you can find a path $p_i:[0,1]→X$ from $x_0$ to $x_i$. Regarded as a singular $1$-simplex $p_i:[v_0v_1]\to X$ via the identification $0≘ v_0,\ 1≘v_1$ , it has boundary $p_i(1)-p_i(0)=x_i-x_0$. Then the sum $\sum η_ip_i$ has the boundary $c$ as all instances of $x_0$ cancel out since, by hypothesis, $\sum η_i=0.$