On Covering Simplices by Dilations in Dimensions 3 and 4

We propose a conjecture regarding the integrally closedness of lattice polytopes with large lattice lengths.

We demonstrate that a lattice simplex in dimension 3 (resp.

4) with lattice length of at least 2 (resp.

3 and no edge has lattice length 5) can be covered by dilated simplices of the form $sQ$, where integer $s\ge 2$ (resp.

3) and $Q$ is a lattice simplex.

The covering property implies these simplices are integrally closed.

As an application, we obtain a simple criterion for the projective normality of ample line bundles on 3-(resp.

4-) dimensional $\mathbb{Q}$-factorial toric Fano varieties with Picard number one.

Along the way, we discover certain unexpected phenomenon.