A topological space X is contractible iff it is homotopic to a point.
That is, there exists a continuous f:X×[0,1] → X such that ∀x∈X f(x,0)=x and f(x,1)=a (for some fixed point a).
In particular, X is path connected; moreover, it must also be simply connected.
Any convex set (and every star convex set) in Rn is contractible.