Hypercube representation

Question

Is it correct to define hypercube $[0,1]^{n}$ as set which consist of: $V\lambda$, where $V=\{v_{1}, .., v_{2^{n}}\}$ is a set of vertices of the given hypercube, $\lambda \in \mathbb{R}_{+}^{2^{n}}$, s.t. $\lambda_{1} + .. + \lambda_{2^{n}}=1$. If it is incorrect, then is there a way to achieve similar representation?

Answer 1

The result you are wanting to show is that the hypercube $H_n:=[0,1]^n$ is the convex hull of the set $V_n := H_n\cap\mathbb Z_+^n$, where $Z_+$ denotes the set of nonnegative integers. It is indeed true, as we may show by induction. For $n=1$, it is clear that $$[0,1] = \left\{\lambda\cdot 0 + (1-\lambda)\cdot 1:0\leqslant \lambda\leqslant 1\right\} = \operatorname{Conv}(\{0,1\}). $$ Now suppose $H_k=\operatorname{Conv}(V_k)$ for some $n\geqslant1$. Since $$V_{n+1} = V_n\times\{0,1\}, $$ it follows that \begin{align} \operatorname{Conv}(V_{n+1}) &= \operatorname{Conv}\left(V_n\times\{0,1\}\right)\\ &= \operatorname{Conv}(V_n)\times\operatorname{Conv}(\{0,1\})\\ &= H_n\times[0,1]\\ &= H_{n+1}. \end{align}