Given a polytope represented by $$Ax\leq b$$ where $A\in\Bbb Z^{m\times n}$, $x\in\Bbb Z^n$ and $b\in\Bbb Z^m$ with $N\geq 0$ number of integer points can we construct a polytope with $N+1$ integer points, $\leq m+c$ constraints and $\leq n+c$ dimensions at a fixed $c\geq0$ in $O(n+m)$ time? What is the best that is known?

Essentially something like this should work. Add an additional dimension $x_{n+1}$ and consider polytope $-1<x_1,\dots,x_n<1$ polytope with exactly one integer point at origin in $n$ dimensions. We want a polytope that at $x_{n+1}=1$ is the first polytope and at $x_{n+1}=0$ is second one. How does one formalize this projective construction by using only linear inequalities without using quadratic inequalities?

In more generality I was looking for an $O(m_1+m_2+n_1+n_2)$ time scheme to construct a polytope $\tilde A\tilde x\leq \tilde b$ with $N_1+N_2$ integer points from polytopes $A_ix_i\leq b_i$ where $A_i\in\Bbb Z^{m_i\times n_i}$, $x_i\in\Bbb Z^{n_i}$ and $b_i\in\Bbb Z^{m_i}$ with $N_i\geq 0$ number of integer points at $i\in\{1,2\}$?

It certainly looks like it could be done with mapping quadratic constraints to linear constraints. However how to go about it is unclear.