Given points $v_1,\dots,v_n\in\mathbb Z^n$ in codimesion $1$ hyperplane $x_1+\dots+x_n=t$ with $0\leq x_{i}$ and a cyclic shift permutation $\sigma$ where

$v_1,\dots,v_n$ when written as columns of a matrix has rank $n$ and each row/column sum same

if $v_i=(x_{i1},\dots,x_{in})$ and $v_j=(x_{j1},\dots,x_{jn})$ then $v_j=(x_{i\sigma(1)},\dots,x_{i\sigma(n)})$

if $v_i=(x_{i1},\dots,x_{in})$ then $\forall i,r,r'\in\{1,\dots,n\}$ with $r\neq r'$ we have $x_{ir}\neq x_{ir'}$

is it possible to characterize the half-space representation of the convex hull of the points efficiently and may be explicitly?

For example for the standard simplex with $n$ points in codimenion $1$ hyperplane $x_1+\dots+x_n=1$ has full rank matrix when corner points are written as matrix and satisfies above properties 1. and 2..