# Linear functions in Lenstra's algorithm

I'm working on implementing Lenstra's algorithm. At the bottom of p.5 (at "construct $$n+1$$ linear functions"), he says to constrain each $$g_i:\mathbb{R}^n\to\mathbb{R}$$ by its value on each of $$n+1$$ vectors (namely, $$v_0,\dots,v_n$$). My question pertains to what Lenstra does after optimizing on each $$g_i$$ (eq.(16)'s paragraph).
Why should replacing $$v_i$$ with $$x$$ satisfying eq.(16) (that is, $$\lvert g_i(x-v_j)\rvert>\frac{3}{2}\lvert g_i(v_i-v_j\rvert$$) cause $$vol(v_0,\dots,v_n)=\det(v_i-v_0)_{i\in[n]}$$ to increase by a factor of $$\frac{3}{2}$$? I guess I'm not seeing exactly what about this choice of $$g_i$$ makes it so desirable to optimize over.
Is there a determinant identity that I'm missing that solves this? It seems that were $$v_i$$ to be replaced by $$v_i=x$$, the angles between some vectors could become quite slim, causing the determinant to perhaps even decrease.