A Solution to an Extreme Value Problem in Discrete Geometry

Subdivide n -dimensional space E n into disjoint n -dimensional unit cubes, and let the set of these cubes be Λ n . The vertices of each cube are called nodes. Let L PT be the set of allcontinuous curve passing through node P(s 1 ,s 2 ,… ,s n ) and node T(t 1 ,t 2 ,… ,t n ) . ∀Γ ∊ LpT, α ∊Λ n define the characteristic function of Γ to oc , χ Γ ( α ) = { 0 , when μ ( α ∩ Γ ) = 0 , 1 , when μ ( α ∩ Γ ) > 0 , Where μ(α ∩ Γ) is Lebesgue measure of point set α ∩ Γ. Name all its non-empty element as α Γ(1) , α Γ(2) , …,α Γ( r (Γ)) , and let λ k ( α Γ( k )) be the weighting coefficients of α Γ( k ) ( k =1,2…, r (Γ)) on Γ . This paper investigated the value of Γ ∈ L P T inf { Σ k = 1 r ( Γ ) λ k ( α Γ ( k ) ) χ ( α Γ ( k ) ) } , and by using discrete mathematics theory and construction method, proved that if all weighting coefficients are equal to 1, then max { m 1 , m 2 , ⋯ m n } ≤ Γ ∈ L P T inf { Σ k = 1 r ( Γ ) λ k ( α Γ ( k ) ) χ ( α Γ ( k ) ) } ≤ Σ k = 1 n ( − 1 ) k − 1 Σ 1 ≤ i 1 < i 2 < ⋯ < i k ≤ n gcd ( m i 1 , m i 2 , ⋯ m i k ) Where m i = | s i − t i | , i = 1,2, …, n . The paper also generalizes the result to a wedge-shaped case.