Some remarks on energy inequalities for harmonic maps with potential

We call the ({delta})-vector of an integral convex polytope of dimension d flat if the ({delta})-vector is of the form ({(1,0,ldots,0,a,ldots,a,0,ldots,0)}), where ({a geq 1}). In this paper, we give the complete characterization of possible flat ({delta})-vectors. Moreover, for an integral convex polytope ({mathcal{P}subset mathbb{R}^N}) of dimension d, we let ({i(mathcal{P},n)=|nmathcal{P}cap mathbb{Z}^N|}) and ({i^*(mathcal{P},n)=|n(mathcal{P} {setminus}partial mathcal{P})cap mathbb{Z}^N|}). By this characterization, we show that for any ({d geq 1}) and for any ({k,ell geq 0}) with ({k+ell leq d-1}), there exist integral convex polytopes ({mathcal{P}}) and ({mathcal{Q}}) of dimension d such that (i) For ({t=1,ldots,k}), we have ({i(mathcal{P},t)=i(mathcal{Q},t),}) (ii) For ({t=1,ldots,ell}), we have ({i^*(mathcal{P},t)=i^*(mathcal{Q},t)}), and (iii) ({i(mathcal{P},k+1) neq i(mathcal{Q},k+1)}) and ({i^*(mathcal{P},ell+1)neq i^*(mathcal{Q},ell+1)}).