A discrete structure with the minimal set

It is known that if the set of nonnegative integral vectors a 1 , … , a m generates the standard lattice Z n then there exists an integral vector g in mon ( a 1 , … , a m ) = { ∑ i = 1 m λ i a i : λ i ∈ Z + , i = 1 , … , m } , such that each integral vector in g + cone ( a 1 , … , a m ) , where cone ( a 1 , … , a m ) = { ∑ i = 1 m λ i a i : λ i ∈ R + , i = 1 , … , m } , is a nonnegative integer linear combination of a 1 , … , a m . Such g is called a swelling-point. Here we are concerned with finding for n ⩾ 2 and m ⩾ n + 1 a finite set K, minimal with respect to set inclusion, such that ( K + cone ( a 1 , … , a m ) ) ∩ Z n is equal to the set of all swelling-points of mon ( a 1 , … , a m ) .