Revista Colombiana de Matemáticas Volumen 31 (1997), páginas 99–108

Some results on the integral geometry of unions of independent families

Paolo Dulio

Politecnico di Milano, ITALIA

Abstract. A notion of independence for families of varieties is presented, and some results of integral geometry are established in relation to their unions. Key words and phrases. Family of varieties, maximal group of invariance of a family of varieties, systems of families of varieties, infinitesimal transformations, Deltheil systems, density of a measurable family of varieties. 1991 AMS. Subject Classification. Primary 60D05. Secondary 52A22.

1. Independent families

Let F_{t} and F_{m} be two families of varieties, p and s dimensional, placed in a space X_{n} and depending on parameters A_{1}, A_{2}, …, A_{t} and B_{1}, B_{2}, …, B_{m} respectively:

F_{t} : F^{λ}(x_{1}, x_{2}, …, x_{n}, A_{1}, A_{2}, …, A_{t}) = 0, λ = 1, 2, …, n − p,

and

F_{m} : G^{μ}(x_{1}, x_{2}, …, x_{n}, B_{1}, B_{2}, …, B_{m}) = 0, μ = 1, 2, …, n − s.

We denote such families by F_{t}(\underline{A}) and F_{m}(\underline{B}). Definition 1.1. The families F_{t}(\underline{A}) and F_{m}(\underline{B}) are said to be independent if there exists no relation φ(\underline{A}, \underline{B}) = 0 between their parameter sets \{\underline{A}\} and \{\underline{B}\}.

Let F_{q} be the family of systems of two independent families F_{t}(\underline{A}) and F_{m}(\underline{B}). The above definition means that the parameters A_{1}, A_{2}, …, A_{t}, B_{1}, B_{2}, …, B_{m} are all essential in F_{q}(\underline{A}, \underline{B}). Further, if the maximal group of invariance of F_{q} is the intersection of the maximal groups of invariance of F_{t} and F_{m}, then F_{q} is also called the independent union of F_{t} and F_{m}.

Hence, every group of invariance of an independent union of independent families is a group of invariance of both of them.

Theorem 1.2. Let G_{1}, G_{2} and G_{3} be the maximal groups of invariance of the families of varieties F_{q}, F_{t} and F_{m}, respectively, where F_{q} = F_{t} + F_{m} is the family of systems of F_{t} and F_{m}. If τ ∈ G_{1}, then τ, or at least τ^{2}, belongs to G_{2} ∩ G_{3}.

Proof. An element of F_{q} is a pair (V, W) where V ∈ F_{t}, W ∈ F_{m}. If τ ∈ G_{1} then τ(V, W) = (τ(V), τ(W)) = (V_{1}, W_{1}) with V_{1} ∈ F_{t} and W_{1} ∈ F_{m}. Three cases may arise:

1. τ(V) = V_{1} and τ(W) = W_{1}. This implies τ ∈ G_{2} and τ ∈ G_{3}, so τ ∈ G_{2} ∩ G_{3}.

2. τ(V) = W_{1}, τ(W) = V_{1}, and τ(V_{1}) ∈ F_{t}. This implies τ(W_{1}) ∈ F_{m}. Consequently τ changes the variety V_{1} ∈ F_{t} into another variety of F_{t}, and therefore τ ∈ G_{2}. Analogously τ changes the variety W_{1} ∈ F_{m} into a variety of F_{m}, and so τ ∈ G_{3}. Hence τ ∈ G_{2} ∩ G_{3}.

3. τ(V) = W_{1}, τ(W) = V_{1}, and τ(V_{1}) ∈ F_{m}. This implies τ(W_{1}) ∈ F_{t}. Consequently τ^{2} changes the variety W ∈ F_{m} into another variety of F_{m}, so that τ^{2} ∈ G_{3}. Analogously, τ^{2} changes the variety V ∈ F_{t} into a variety of F_{t}, so τ^{2} ∈ G_{2}. Hence τ^{2} ∈ G_{2} ∩ G_{3}. □ ✓

Remark 1.3. Clearly the maximal group of invariance G_{1} of F_{q} = F_{t} + F_{m} always contains the intersection G_{2} ∩ G_{3} of the maximal groups of invariance of F_{t} and F_{m}. Therefore, in the cases 1 and 2 of the above theorem, we have G_{1} = G_{2} ∩ G_{3}, so that, if F_{t} and F_{m} are independent, F_{q} is their independent union.

2. Systems of independent families

Let F_{q}(\underline{A}, \underline{B}) = F_{t}(\underline{A}) + F_{m}(\underline{B}), (q = t + m) be the family of systems of two independent families. Let us suppose that F_{t} and F_{m} are both measurable with respect to the maximal group of invariance G_{r} of F_{q}(\underline{A}, \underline{B}), r ≥ q, with respective densities

dψ_{t} = Φ_{1}(A_{1}, A_{2}, …, A_{t})dA_{1} ∧ dA_{2} ∧ ⋯ ∧ dA_{t},  (1)

and

dψ_{m} = Φ_{2}(B_{1}, B_{2}, …, B_{m})dB_{1} ∧ dB_{2} ∧ ⋯ ∧ dB_{m}.  (2)

ON THE INTEGRAL GEOMETRY OF UNIONS OF INDEPENDENT FAMILIES 101

The group H_{r} associated to G_{r} in the parameter space X_{q} is measurable if and only if there exists a single non-trivial solution Φ = Φ(C_{1}, C_{2}, …, C_{q}) of the Deltheil system

∑_{i=1}^{q} \frac{∂(ξ_{h}^{i}Φ)}{∂C_{i}} = 0, h = 1, 2, …, r.  (3)

Here ξ_{h}^{i} are the coefficients of the infinitesimal transformations of the group H_{r} and C_{1}, C_{2}, …, C_{q}, the essential parameters of F_{q} ([2], [7]).

Since F_{t} and F_{m} are independent families, the Deltheil system (3) may be written in the form

∑_{k=1}^{t} \frac{∂(ξ_{h}^{k}Φ)}{∂A_{k}} + ∑_{j=1}^{m} \frac{∂(ξ_{h}^{j+t}Φ)}{∂B_{j}} = 0, h = 1, 2, …, r.  (4)

The coefficients of the infinitesimal transformations ([7]) are

ξ_{h}^{i} = (\frac{∂C′_{i}}{∂α_{h}} ) |_{0},

where {C′_{1}, C′_{2}, …, C′_{q}} is a group isomorphic to G_{r} and α_{1}, α_{2}, …, α_{r} are the parameters of G_{r}. The independence of F_{t} and F_{m} also implies that

{C′_{1}, C′_{2}, …, C′_{q}} = {A′_{1}, A′_{2}, …, A′_{t}, B′_{1}, B′_{2}, …, B′_{m}},

where {A′_{1}, A′_{2}, …, A′_{t}} is the contribution of F_{t} and {B′_{1}, B′_{2}, …, B′_{m}} that of F_{m}. That is, {A′_{1}, A′_{2}, …, A′_{t}} and {B′_{1}, B′_{2}, …, B′_{m}} jointly determine a group isomorphic to G_{r} and are associated to the families F_{t} and F_{m}, respectively. Consequently, we have

ξ_{h}^{k} = (\frac{∂A′_{k}}{∂α_{h}} ) |_{0}, k = 1, 2, …, t,

ξ_{h}^{j+t} = (\frac{∂B′_{j}}{∂α_{h}} ) |_{0}, j = 1, 2, …, m.

Since F_{t} is measurable with respect to G_{r} then by (1) we obtain

∑_{k=1}^{t} \frac{∂(ξ_{h}^{k}Φ_{1}(A_{1}, A_{2}, …, A_{t}))}{∂A_{k}} = 0, h = 1, 2, …, r,  (5)

and since also F_{m} is measurable with respect to G_{r}, it follows from (2) that

∑_{j=1}^{m} \frac{∂(ξ_{h}^{j+t}Φ_{2}(B_{1}, B_{2}, …, B_{m}))}{∂B_{j}} = 0, h = 1, 2, …, r.  (6)

Now take Φ = Φ_{1}Φ_{2} and replace in (3). We get

∑_{i=1}^{q} \frac{∂(ξ_{h}^{i}Φ)}{∂C_{i}} = ∑_{k=1}^{t} \frac{∂(ξ_{h}^{k}Φ)}{∂A_{k}} + ∑_{j=1}^{m} \frac{∂(ξ_{h}^{j+t}Φ)}{∂B_{j}}

= Φ_{2}(B_{1}, B_{2}, …, B_{m}) ∑_{k=1}^{t} \frac{∂(ξ_{h}^{k}Φ_{1}(A_{1}, A_{2}, …, A_{t}))}{∂A_{k}}  (7)

+ Φ_{1}(A_{1}, A_{2}, …, A_{t}) ∑_{j=1}^{m} \frac{∂(ξ_{h}^{j+t}Φ_{2}(B_{1}, B_{2}, …, B_{m}))}{∂B_{j}} = 0.

The above argument shows that the Deltheil system associated to the group H_{r}, which is isomorphic to the maximal group of invariance G_{r} of the family of systems of two independent families, both measurable with respect to G_{r}, always has at least a non-trivial solution. Furthermore, (7) ensures the following result.

Theorem 2.1. If F_{q} = F_{t} + F_{m} is the family of systems of two independent families of varieties F_{t}, F_{m} (q = t + m), both measurable with respect to the maximal group of invariance G_{r} of F_{q} (r ≥ q), the densities being dψ_{t} and dψ_{m}, then F_{q} assumes, with respect to G_{r}, the density dψ_{q} = dψ_{t} ∧ dψ_{m}.

Corollary 2.2. If the group H_{r} isomorphic to G_{r} in the parameter space X_{q} of F_{q} is transitive, then F_{q} is measurable, and its density is dψ_{q} = dψ_{t} ∧ dψ_{m}.

Remark 2.3. The assumption that F_{t} and F_{m} are both measurable with respect to the maximal group of invariance G_{r} of F_{q} implies that G_{r} is a group of invariance of both these families. Consequently, by Theorem 1.2, F_{q} is the independent union of F_{t} and F_{m}.

More in general, the theorem holds whenever F_{t} and F_{m} have a density with respect to a same group of invariance.

Example 2.4. In the projective space ℙ_{3}, let F_{9} be the family of nondegenerate quadrics having elliptic points (Δ < 0, where Δ is the determinant of the quadrics), and let F_{6} be the family of pairs plane–point, with the point out of the plane:

F_{9} : x^{2} + (A^{2} + C)y^{2} + (B + D)z^{2} + 2Axy + 2Bxz

+ 2(AB + E)yz + 2Lx + 2My + 2Nz + P = 0,

F_{6} : \{ \begin{matrix}A_{1}x + A_{2}y + A_{3}z + 1 = 0, \\ x = α_{1}, \\ y = α_{2}, \\ z = α_{3}, (∑_{i=1}^{3} A_{i}α_{i} + 1 ≠ 0) . \end{matrix}

These families are measurable ([3], [5]), with respective densities

dψ_{9} = |Δ|^{− \frac{5}{2}} dA ∧ dB ∧ dC ∧ dD ∧ dE ∧ dL ∧ dM ∧ dN ∧ dP,

and

dψ_{6} = (A_{1}α_{1} + A_{2}α_{2} + A_{3}α_{3} + 1)^{−4}dA_{1} ∧ dA_{2} ∧ dA_{3} ∧ dα_{1} ∧ dα_{2} ∧ dα_{3}.

The maximal group of invariance of the family of systems F_{15} = F_{9} + F_{6} is the projective group G_{15}, with respect to which both F_{9} and F_{6} are measurable. Then F_{15} is the independent union of F_{9} and F_{6}, and so, by Theorem 2.1, it assumes the density dψ_{15} = dψ_{9} ∧ dψ_{6} with respect to G_{15}. Here r = q = 15, so that if the determinant \overline{Δ} = |ξ_{h}^{i}| ≠ 0, then Corollary 2.2 holds. Note that \overline{Δ} is a polynomial; hence, it is non-trivial as soon as we can find a point Q ∈ X_{15} whose coordinates specify a variety V ∈ F_{15} such that \overline{Δ}(Q) ≠ 0. If we take the point

Q = (A, B, C, D, E, L, M, N, P, A_{1}, A_{2}, A_{3}, α_{1}, α_{2}, α_{3})
= (1, 1, 2, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1)

then Δ = −2 and A_{1}α_{1} + A_{2}α_{2} + A_{3}α_{3} + 1 = 4. Therefore, the variety V associated to Q belongs to F_{15}. Moreover, we have \overline{Δ}(Q) = −3072 ≠ 0. Hence, we can apply Corollary 2.2, to deduce that F_{15} is measurable, with density

dψ_{15} = |Δ|^{− \frac{5}{2}} (∑_{i=1}^{3}(A_{i}α_{i} + 1))^{−4} dA ∧ ⋯ ∧ dP ∧ dA_{1} ∧ ⋯ ∧ dα_{3}.

In the same way we can handle the independent union \overline{F}_{15} = \overline{F}_{9} + F_{6} of the measurable families [3]

\overline{F}_{9} : x^{2} + (A^{2} + C)y^{2} + (B + D)z^{2} + 2Axy + 2Bxz
+ 2(AB + E)yz + 2Lx + 2My + 2Nz + P = 0

consisting of non-degenerate quadrics having hyperbolic points (C(DC −E^{2}) ≤ 0 and Δ > 0, where Δ is the determinant of the quadrics) and the previous F_{6}. Here we can take the point

Q = (A, B, C, D, E, L, M, N, P, A_{1}, A_{2}, A_{3}, α_{1}, α_{2}, α_{3})
= (1, 1, 2, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1).

With this choice we have Δ = 1 and A_{1}α_{1} + A_{2}α_{2} + A_{3}α_{3} + 1 = 4. Therefore the variety V associated to Q belongs to \overline{F}_{15}. Moreover \overline{Δ}(Q) = −384 ≠ 0 and hence, by [3] and Corollary 2.2, \overline{F}_{15} is measurable, with density

d\overline{ψ}_{15} = |Δ|^{− \frac{5}{2}} (∑_{i=1}^{3}(A_{i}α_{i} + 1))^{−4} dA ∧ ⋯ ∧ dP ∧ dA_{1} ∧ ⋯ ∧ dα_{3}.

Remark 2.5. If H_{r} is not transitive, the family F_{q} is not measurable, even though r ≥ q and the Deltheil system associated to H_{r} have a solution, as this solution is not unique. Here we have an unusual kind of non-measurability, different from that pointed out by the Stoka’s second condition [8], since in this case the family F_{q} has different measures with respect to the same group, namely, the maximal group of invariance.

3. Iterated unions on a family of varieties

A special kind of independent union of families of varieties can be obtained by replacing the parameters A_{1}, A_{2}, …, A_{t} of a family F_{t} by new parameters B_{1}, B_{2}, …, B_{t} having no relation with the former, and by taking F_{q}(\underline{A}, \underline{B}) = F_{t}(\underline{A}) + F_{t}(\underline{B}). In this case we write F_{q} = 2F_{t}. We can easily see that the groups of invariance of F_{q} and F_{t} are the same. Hence F_{q} actually is an independent union. By iterating this construction we can build up the family F_{q}(\underline{A}^{1}, \underline{A}^{2}, …, \underline{A}^{m}) = mF_{t} that is the independent union of m copies of the family F_{t}.

Let F_{q} = 2F_{t} and assume F_{t} to be measurable with respect to the maximal group of invariance G_{r}, r ≥ 2t (so that F_{t} is measurable), with density

ψ_{t} = ϕ_{t}(A_{1}, A_{2}, …, A_{t})dA_{1} ∧ dA_{2} ∧ ⋯ ∧ dA_{t}.

Further assume that the coefficients of the infinitesimal transformations ξ_{h}^{k}, k = 1, 2, …, t; h = 1, 2, …, r, are polynomials of degree d ≤ 1, that is

ξ_{h}^{k} = f_{h}^{k}(A_{1}, A_{2}, …, A_{t}) = ∑_{j=1}^{t} λ_{hj}^{k}A_{j} + μ_{h}^{k}, λ_{hj}^{k}, μ_{h}^{k} ∈ ℝ.  (8)

The Deltheil system associated to F_{q} is

∑_{i=1}^{q} \frac{∂(ξ_{h}^{i}Φ)}{∂C_{i}} = ∑_{k=1}^{t} \frac{∂(ξ_{h}^{k}Φ)}{∂A_{k}} + ∑_{k=1}^{t} \frac{∂(η_{h}^{k}Φ)}{∂B_{k}}, h = 1, 2, …, r,  (9)

where η_{h}^{k} = f_{h}^{k}(B_{1}, B_{2}, …, B_{t}).

Now let Φ = [Φ_{t}(A_{1}, A_{2}, …, A_{t})]^{α}[Φ_{t}(B_{1}, B_{2}, …, B_{t})]^{2−α}. Replacing in (9) we get

∑_{k=1}^{t} \frac{∂ [ξ_{h}^{k}Φ_{t}(\underline{A})^{α}Φ_{t}(\underline{B})^{2−α}]}{∂A_{k}} + ∑_{k=1}^{t} \frac{∂ [η_{h}^{k}Φ_{t}(\underline{A})^{α}Φ_{t}(\underline{B})^{2−α}]}{∂B_{k}} =

ON THE INTEGRAL GEOMETRY OF UNIONS OF INDEPENDENT FAMILIES 105

= ∑_{k=1}^{t} [\frac{∂ξ_{h}^{k}}{∂A_{k}} Φ_{t}(\underline{A})^{α}Φ_{t}(\underline{B})^{2−α} + ξ_{h}^{k}αΦ_{t}(\underline{A})^{α−1} \frac{∂Φ_{t}(\underline{A})}{∂A_{k}}Φ_{t}(\underline{B})^{2−α}+

+\frac{∂η_{h}^{k}}{∂B_{k}} Φ_{t}(\underline{A})^{α}Φ_{t}(\underline{B})^{2−α} + η_{h}^{k}(2 − α)Φ_{t}(\underline{A})^{α}Φ_{t}(\underline{B})^{1−α} \frac{∂Φ_{t}(\underline{B})}{∂B_{k}}]

= Φ_{t}(\underline{A})^{α}Φ_{t}(\underline{B})^{2−α} ∑_{k=1}^{t} [\frac{∂f_{h}^{k}(\underline{A})}{∂A_{k}} + αf_{h}^{k}(\underline{A})Φ_{t}(\underline{A})^{−1} \frac{∂Φ_{t}(\underline{A})}{∂A_{k}}+

+ \frac{∂f_{h}^{k}(\underline{B})}{∂B_{k}} + (2 − α)f_{h}^{k}(\underline{B})Φ_{t}(\underline{B})^{−1} \frac{∂Φ_{t}(\underline{B})}{∂B_{k}}] .

(10)

Since F_{t} is measurable, we have

∑_{k=1}^{t} [f_{h}^{k}(\underline{A})\frac{∂Φ_{t}(\underline{A})}{∂A_{k}} + \frac{∂f_{h}^{k}(\underline{A})}{∂A_{k}}Φ_{t}(\underline{A})] = 0,  (11)

and

∑_{k=1}^{t} [f_{h}^{k}(\underline{B})\frac{∂Φ_{t}(\underline{B})}{∂B_{k}} + \frac{∂f_{h}^{k}(\underline{B})}{∂B_{k}}Φ_{t}(\underline{B})] = 0.  (12)

By (11) and (12) we have

∑_{k=1}^{t} \frac{∂f_{h}^{k}(\underline{A})}{∂A_{k}} = − ∑_{k=1}^{t} f_{h}^{k}(\underline{A}) [Φ_{t}(\underline{A})]^{−1} \frac{∂Φ_{t}(\underline{A})}{∂A_{k}},  (13)

and

∑_{k=1}^{t} \frac{∂f_{h}^{k}(\underline{B})}{∂B_{k}} = − ∑_{k=1}^{t} f_{h}^{k}(\underline{B}) [Φ_{t}(\underline{B})]^{−1} \frac{∂Φ_{t}(\underline{B})}{∂B_{k}}.  (14)

Moreover, from (8) we obtain

\frac{∂f_{h}^{k}(\underline{A})}{∂A_{k}} = \frac{∂f_{h}^{k}(\underline{B})}{∂B_{k}} = λ_{hk}^{k}.  (15)

Therefore, by replacing (13), (14) and (15) in (10), we get

Φ_{t}(\underline{A})^{α}Φ_{t}(\underline{B})^{2−α} ∑_{k=1}^{t} [\frac{∂f_{h}^{k}(\underline{A})}{∂A_{k}} − α\frac{∂f_{h}^{k}(\underline{A})}{∂A_{k}} + \frac{∂f_{h}^{k}(\underline{B})}{∂B_{k}} − (2 − α)\frac{∂f_{h}^{k}(\underline{B})}{∂B_{k}}]

= (α − 1)Φ_{t}(\underline{A})^{α}Φ_{t}(\underline{B})^{2−α} ∑_{k=1}^{t} [\frac{∂f_{h}^{k}(\underline{B})}{∂B_{k}} − \frac{∂f_{h}^{k}(\underline{A})}{∂A_{k}}] = 0.

Consequently, the following result holds.

Theorem 3.1. Let F_{t} = F_{t}(A_{1}, A_{2}, …, A_{t}) be a family of varieties which is measurable with respect to its maximal group of invariance G_{r}, with density

Φ_{t}(A_{1}, A_{2}, …, A_{t})dA_{1} ∧ dA_{2} ∧ ⋯ ∧ dA_{t}.

If the coefficients of the infinitesimal transformations are polynomials of degree d ≤ 1, then the independent union F_{q} = 2F_{t}, r ≥ q = 2t, is a non-trivially non-measurable family, and it assumes at least the densities

[Φ_{t}(A_{1}, A_{2}, …, A_{t})]^{α} [Φ_{t}(B_{1}, B_{2}, …, B_{t})]^{2−α} dA_{1} ∧ ⋯ ∧ dA_{t} ∧ dB_{1} ∧ ⋯ ∧ dB_{t},

for every α ∈ ℝ.

Remark 3.2. A family F_{q} is said to be a trivially non-measurable family of varieties, if its maximal group of invariance G_{r} depends on r < q parameters. In this case the group H_{r} isomorphic to G_{r} is not transitive ([1], [2]).

By an analogous argument we can also state the following theorem.

Theorem 3.3. Let F_{t} = F_{t}(A_{1}, A_{2}, …, A_{t}) be a family of varieties which is measurable with respect to its maximal group of invariance G_{r}, with density

Φ_{t}(A_{1}, A_{2}, …, A_{t})dA_{1} ∧ dA_{2} ∧ ⋯ ∧ dA_{t}.

If the coefficients of the infinitesimal transformations are polynomials of degree d ≤ 1, then the independent union F_{q} = mF_{t}, r ≥ q = mt, is, for every integer m ≠ 1, a non-trivially non-measurable family, and it assumes at least the densities

[Φ_{t}(\underline{A}^{1})]^{α_{1}} [Φ_{t}(\underline{A}^{2})]^{α_{2}} ⋯ [Φ_{t}(\underline{A}^{m})]^{α_{m}} ∧^{t} d\underline{A}^{1} ∧^{t} d\underline{A}^{2} ∧^{t} ⋯ ∧^{t} d\underline{A}^{m},

where \underline{A}^{i} = (A_{1}^{i}, A_{2}^{i}, …, A_{t}^{i}), i = 1, 2, …, m, is the set of parameters of the i-th copy of F_{t}, ∧^{t}d\underline{A}^{i} = dA_{1}^{i} ∧ dA_{2}^{i} ∧ ⋯ ∧ dA_{t}^{i}, and α_{1}, α_{2}, …, α_{m} are real numbers such that α_{1} + α_{2} + ⋯ + α_{m} = m.

Example 3.4. From [11] we know that the family F_{n+1} of hyperspheres of the n-dimensional projective space ℙ_{n}

x_{1}^{2} + x_{2}^{2} + ⋯ + x_{n}^{2} − 2u_{1}x_{1} − 2u_{2}x_{2} − ⋯ − 2u_{n}x_{n} + v = 0,

is measurable with respect to its maximal group of invariance, namely the group of similarities G_{\frac{n(n+1)}{2} +1}(ρ, \underline{α}, \underline{θ}).

Here ρ, \underline{α} = \{α_{1}, α_{2}, …, α_{n}\} and

\underline{θ} = \{θ_{12}, …, θ_{1n}, θ_{23}, …, θ_{2n}, …, θ_{(n−1)n}\}

are, respectively, the homothety, translation and rotation parameters. The density is

dψ_{n+1} = R^{−(n+1)}du_{1} ∧ du_{2} ∧ ⋯ ∧ du_{n} ∧ dR,

where

R = [(∑_{j=1}^{n} u_{j}^{2}) − v]^{− \frac{1}{2}}

is the radius of the hyperspheres. The coefficients of the infinitesimal transformations are ([11])

(\frac{∂u′_{j}}{∂ρ} ) |_{0} = −u_{j}, j = 1, 2, …, n,

(\frac{∂u′_{j}}{∂α_{i}} ) |_{0} = \{ \begin{matrix}0, & if j ≠ i, \\ −1, & if j = i, j, i = 1, 2, …, n, \end{matrix}

(\frac{∂u′_{j}}{∂θ_{hk}} ) |_{0} = \{ \begin{matrix}0, & if j ≠ h, k, \\ u_{k}, & if j = k, \\ −u_{h}, & if j = h, j = 1, 2, …, n, h < k = 1, 2, …, n, \end{matrix}

(\frac{∂v′}{∂ρ} ) |_{0} = −2v,

(\frac{∂v′}{∂α_{i}} ) |_{0} = −2u_{i}, i = 1, 2, …, n,

(\frac{∂v′}{∂θ_{hk}} ) |_{0} = 0, h < k = 1, 2, …, n.

Note that all of them are polynomials of degree d ≤ 1.

Let m, n be two integers such that

m(n + 1) ≤ \frac{n(n + 1)}{2} + 1.

For fixed m this holds for any n ≥ n_{0}, where n_{0} is the least integer greater than

\frac{2m + 1 + \sqrt{4m^{2} + 4m − 7}}{2},

so that n ≥ 2m. By Theorem 3.3, the independent union F_{m(n+1)} = mF_{n+1},

(x_{1} − u_{11})^{2} + (x_{2} − u_{12})^{2}+ ⋯ + (x_{n} − u_{1n}^{2}) = R_{1}^{2},
(x_{1} − u_{21})^{2} + (x_{2} − u_{22})^{2}+ ⋯ + (x_{n} − u_{2n}^{2}) = R_{2}^{2},
⋮
(x_{1} − u_{m1})^{2} + (x_{2} − u_{m2})^{2}+ ⋯ + (x_{n} − u_{mn}^{2}) = R_{m}^{2},

is, for every integer m ≥ 2, a non-trivially non-measurable family of varieties of the projective space ℙ_{2m}. This family assumes, at least, the densities

[R_{1}^{a_{1}}R_{2}^{a_{2}} ⋯ R_{m}^{a_{m}}]^{−(n+1)} du_{11} ∧ ⋯ ∧ du_{mn} ∧ dR_{1} ∧ ⋯ ∧ dR_{m}, ∑_{s=1}^{m} a_{s} = m.

If a_{1} = a_{2} = ⋯ = a_{m} = 1, this yields the product measure.

References

[1] L. Bianchi, Lezioni sulla teoria dei gruppi finiti di trasformazioni, N. Zanichelli, Bologna, 1928.

[2] R. Deltheil, Probabilités géométriques, Gauthier-Villars, Paris, 1926.

[3] V. Pipitone & G. Russo, Misurabilitá delle famiglie di quadriche di tipo iperbolico e di tipo ellittico dello spazio proiettivo ℙ_{3}, Atti Acc. Sc. Lett. ed Arti di Palermo (4) 36-(1) (1976-77), 445–448.

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[8] M. I. Stoka, Geometria Integrale in uno spazio euclideo 𝔼_{n}, Boll. Un. Mat. Ital. (3) 13 no. 4 (1958), 470–485.

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[11] M. I. Stoka, La misurabilitá della famiglia delle ipersfere nello spazio proiettivo ℙ_{n}, Atti Acc. Sc. Lett. ed Arti di Palermo (4) 36 no. 1 (1976-77), 511–516.

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(Recibido en mayo de 1996; revisado en noviembre de 1997)

Dipartimento di Matematica

Politecnico di Milano Piazza Leonardo da Vinci 32,

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