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(Positively) Homogeneous Function F on a Compact Domain Ω Rn Reduces to Integrating a Related Function on the ⊂ Boundary ∂Ω

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(Positively) Homogeneous Function F on a Compact Domain Ω Rn Reduces to Integrating a Related Function on the ⊂ Boundary ∂Ω PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 3, March 1999, Pages 813{818 S 0002-9939(99)04930-8 INTEGRATION AND HOMOGENEOUS FUNCTIONS JEAN B. LASSERRE (Communicated by David H. Sharp) Abstract. We show that integrating a (positively) homogeneous function f on a compact domain Ω Rn reduces to integrating a related function on the ⊂ boundary @Ω. The formula simplifies when the boundary @Ω is determined by homogeneous functions. Similar results are also presented for integration of exponentials and logarithms of homogeneous functions. 1. Introduction We consider the integration of a continuous (positively) homogeneous function f : Rn R on a compact domain Ω with boundary @Ω. Using Euler’s identity for homogeneous→ functions, Green’s formula simplifies so that integrating f on Ω reduces to integrating a (simply related) function on the boundary @Ω. In the n particular case where Ω := x R gi(x) ai;i=1;:::;m and where the func- { ∈ | ≤ } tions gi are each (positively) homogeneous of degree pi, the formula is even simpler. Actually, an alternative proof that uses only Euler’s formula is also outlined. We thus extend to more general domains a result in [4] for integrating a homo- geneous function on a convex polytope. A potential application is the integration of arbitrary continuous functions on a compact set Ω. Indeed, as the polynomials are sums of homogeneous polynomials, and are dense in C(Ω), the space of continuous functions on Ω with the sup norm, or even in L1(Ω), one could approximate Ω fdx by i Ω Pidx,wherethePi’s are homogeneous polynomials, and therefore use the previous result. This result could also be used in finite element methods forR the integrationP R of a poynomial on each individual volume element. Finally, we also provide simple formulas for the integration of exponentials and logarithms of homogeneous functions. For instance, integrating log f on a convex polytope Ω reduces to integrating log f on @Ω and to computing the volume of Ω. 2. Integration of a homogeneous function Let f : Rn R be a real (positively) homogeneous function of degree p (or in short, f is p→-homogeneous), i.e. f(λx)=λpf(x) for all λ>0;x Rn.For a (positively) p-homogeneous function that is continuously differentiable,∈ Euler’s Received by the editors July 8, 1997. 1991 Mathematics Subject Classification. Primary 65D30. Key words and phrases. Numerical integration in Rn, homogeneous functions. c 1999 American Mathematical Society 813 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 814 JEAN B. LASSERRE formula states that (2.1) pf(x)= f(x);x for all x: h∇ i 2.1. On the Riemann-Green formula. We first treat the 2-dimensional case. Let Ω be a compact domain in R2 with boundary @Ω. Let P (x; y)andQ(x; y) be continuously differentiable on Ω. Then, under some regularity conditions, the well-known Riemann-Green formula (cf. [5]) states that @Q @P (2.2) ( )dx dy = Pdx+Qdy: @x − @y ZZΩ Z@Ω This yields: Lemma 2.1. Let f (x; y) be a continuously differentiable p-homogeneous function on Ω.Then (2.3) (p+2) fdxdy= f(x; y)(xdy ydx): − ZZΩ Z@Ω Proof. As f is p-homogeneous, from (2.1) @f @f pf(x; y)=x + y (x; y) Ω: @x @y ∀ ∈ Let P (x; y):= yf(x; y)andQ(x; y):=xf(x; y) for every (x; y) Ω. − ∈ @Q @P @f @f =2f(x; y)+x + y =(p+2)f(x; y) (x; y) Ω: @x − @y @x @y ∀ ∈ Hence, applying the Riemann-Green formula (2.2) to P and Q yields (p +2) fdxdy= f(x; y)(xdy ydx); − ZZΩ Z@Ω the desired result. When f 1 (i.e. p = 0), one immediately retrieves the well-known formula ≡ 1 area(Ω) = xdy ydx: 2 − Z@Ω 2.2. The general case. The previous result generalizes to Rn as follows: Lemma 2.2. Let f : Rn R be a continuously differentiable p-homogeneous func- tion and Ω a compact domain→ Rn with boundary @Ω.Then (2.4) (n+p) fd! = M;~~ n f(M)dσ; h i ZΩ Z@Ω where ~n is the unit outward-pointing normal to @Ω. @ Proof. With notation as in [6], let X be the vector field X := xj . From j @xj Proposition 2.3 in [6], we have P (2.5) div(X)fd!+ Xf d! = X;~n fdσ: h i ZΩ ZΩ Z@Ω Now, div(X)=nand from (2.1), Xf = pf so that the result follows. Thus, integrating f on Ω reduces to integrating M;~~ n f on the boundary @Ω. We next show that Lemma 2.2 further simplifies inh the casei where the boundary @Ω is determined by homogeneous functions. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use INTEGRATION AND HOMOGENEOUS FUNCTIONS 815 3. Boundary determined by homogeneous functions We now consider the special case where n n (3.1) Ω( R ):= x R gi(x) ai;i=1;:::;m ; ⊂ { ∈ | ≤ } and the functions gi are each continuously differentiable (positively) pi-homogeneous functions, i =1;:::;m. We still assume that Ω is compact. Finally, let Ωi := x m { ∈ Ω gi(x)=ai so that @Ω= Ωi. | } i=1 Theorem 3.1. Assume thatSf is a continuously differentiable q-homogeneous func- tion and the functions gi are each continuously differentiable and pi-homogeneous, i =1;:::;m.Then m f (3.2) (n+q) fd! = piai dσ: g i Ω i=1 Ωi Z X Z || ∇ || In particular, with f 1, one obtains ≡ m 1 (3.3) n vol(Ω) = piai g i − dσ: × || ∇ || i=1 Ωi X Z Proof. Lemma2.2appliessothat m (3.4) (n+q) fd! = X;~n fdσ: h i Ω i=1 Ωi Z XZ 1 Now, ~n = g i − g i on Ωi. Therefore, using (2.1), we have on Ωi || ∇ || ∇ X; gi = x; gi(x) = pigi(x)=piai; h ∇ i h ∇ i which yields the desired result. Remark. When Ω is a convex polytope, i.e. gi(x)= Ai;x and pi =1,i=1;:::;m, h i then g i = A i and thus (3.2) simplifies to || ∇ || || || m a (n + q) fd! = i fdσ: A i Ω i=1 Ω i Z X || || Z In addition, one may iterate the process for fdσ (cf. [4] for more details). Ωi An alternative proof. Interestingly enough, thereR is a direct proof that does not use Green’s formula. It was used in [4] for the case where Ω is a convex polytope. pi Write ai = bi ;i=1;:::;m, and let (3.5) h(b):= fd! = fdx: Ω g (x) bpi;i=1;:::;m: Z Z{ i ≤ i } It is immediate that h(λb)=λn+qh(b), i.e. h is (n + q)-homogeneous and continu- ously differentiable. In addition, one may show that @h pi 1 f = pibi − dσ: @bi g i ZΩi || ∇ || Hence, applying Euler’s formula to h yields m f (3.6) (n + q)h(b)= h(b);b = piai dσ; h∇ i g i i=1 Ωi X Z || ∇ || the desired result. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 816 JEAN B. LASSERRE Example. In R2, consider the domain Ω:= (x; y) R2 xy 1; x2 + y2 R2; x; y 0 { ∈ | ≤ ≤ ≥ } and its volume Ω d! (i.e. f 1 is 0-homogeneous). Note that in (3.3), the coordinate≡ axis boundaries of Ω do not contribute since R the boundary values ai are zero in this case. 2 2 Both gi(x; y):=xy and g (x; y):=x +y are 2-homogeneous. Ω := (x; y) 2 1 { ∈ Ω xy =1 and Ω := (x; y) Ω x2 + y2 = R2 . g (M) = x2 + y2 so | } 2 { ∈ | } ||∇ 1 || that on Ω we have g (M) = (x4 +1)=x2 and dσ = (x4 +1)=x4dx,which 1 1 p yields ||∇ || p p a 1 1 (3.7) g 1 − dσ = x− dx =2log(a); || ∇ || 1 ZΩ1 Za− where x = a is the solution to x2 + y2 = R2;xy=1. Similarly, g =2Ron Ω ,sothat || ∇ 2 || 2 1 g − dσ =2 (1=2R) (Rα)=α, || ∇ 2 || × × ZΩ2 2 where α = arctan (a− ). Hence, Theorem 3.1 yields 2 2 2vol(Ω)=4log(a)+R arctan (a− ); 2 2 i.e. vol(Ω) = 2 log (a)+R arctan (a− )=2. This can be retrieved directly. If we now take Ω := (x; y) xy 1; b y a; x 0 with a b>0, we get Ω := (x; y) Ω y = a{ and Ω| :=≤ (x; y≤) Ω≤ y=≥ }b . Therefore,≥ 2 { ∈ | } 3 { ∈ |− − } 1 a 1 1 1 1 g − dσ =log( ); g − dσ = ; g − dσ = ; || ∇ 1 || b || ∇ 2 || a || ∇ 3 || b ZΩ1 ZΩ2 ZΩ3 so that Theorem 3.1 yields a 1 1 a 2vol(Ω) = 2 log ( )+a b =2log( ); b a− b b i.e. vol(Ω) = log (a=b), which is immediate to check. 4. Integrating exponentials and logarithms In this section, we consider exponentials and logarithms of homogeneous func- tions. Indeed, we show that integrating such functions on Ω reduces to integrating a related function on the boundary @Ω. 4.1. Exponentials. We first consider the case of an exponential eh(x) where h : Rn R is q-homogeneous. The particular case where h = c; x was considered in [1]→ and [2] and was used to compute the number of integralh pointsi in a convex polytope. We have the following result.
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