Signed product-formula measure with minimal support

Construct, for every finite root system R ⊂ ℝ^N and nonnegative multiplicity θ ∈ θ(R), a signed measure μ^{R(θ)}_{a_1,a_2} on ℝ^N that represents the product J^{R(θ)}_{a_1}(x) J^{R(θ)}_{a_2}(x) as ∫ J^{R(θ)}_{a}(x) dμ^{R(θ)}_{a_1,a_2}(a) and whose support is exactly contained in the Minkowski sum of the convex hulls of the H(R)-orbits of a_1 and a_2, i.e., conv(H(R) a_1) + conv(H(R) a_2).

Background

Trimeche proved that there exists a signed measure realizing the product formula, supported in a Euclidean ball; the paper improves the geometric understanding of the support (assuming the positivity conjecture) and conjectures that a signed representation can always be chosen with support limited to the sum of convex hulls of the reflection-group orbits.

Achieving such a sharp support would align the signed product-formula representation with the geometric constraint derived in the paper under the assumption of the positive product formula.