Proof that domains of positivity of symmetric nondegenerate bilinear forms are self-dual cones?

Question

Max Koecher (for example, in The Minnesota Notes on Jordan Algebras and Their Applications; new edition: Springer Lecture Notes in Mathematics, number 1710, 1999), defined a domain of positivity for a symmetric nondegenerate bilinear form $B: X \times X \rightarrow \mathbb{R}$ on a finite dimensional real vector space $X$, to be an open set $Y \subseteq X$ such that $B(x,y) > 0$ for all $x,y \in Y$, and such that if $B(x,y) > 0$ for all $y \in Y$, then $x \in Y$. (More succinctly, perhaps, we could say it's a maximal set $Y \subseteq X$ such that $B(Y,Y) > 0$.) Aloys Krieger and Sebastian Walcher, in their notes to chapter 1 of this book, state that "In the language used today, a domain of positivity is a self-dual open proper convex cone." [I now believe this is wrong; see my answer below for what I think is true instead.] It's quite easy to prove that it's an open proper convex cone. (Proper means it contains no nonzero linear subspace of $X$, i.e. that its closure is pointed.) But, although I have a vague recollection of having encountered a proof once in a paper on homogeneous self-dual cones, I haven't succeeded in finding it again, or in supplying it myself. I'm pretty sure Krieger and Walcher's claim is correct—for example, the 1958 paper by Koecher that is generally cited (along with a 1960 paper by Vin'berg) for the proof of the celebrated result that the (closed) finite-dimensional homogeneous self-dual cones are precisely the cones of squares in finite dimensional formally real Jordan algebras, is titled "The Geodesics of Domains of Positivity" (but in German).

The most natural way to prove this would be to find a positive semidefinite nondegenerate $B'$, such that the cone is a domain of positivity for $B'$ as well. In principle, $B'$ might depend on the domain $Y$. (While maximal in the subset ordering, domains of positivity for a given form $B$ are not unique.) But a tempting possibility, independent of $Y$, is to transform to a basis for $X$ in which $B$ is diagonal, with diagonal elements $\pm 1$, change the minus signs to plus signs, and transform back to obtain $B'$.

To clarify the question: we will define a cone $K$ in a real vector space $X$ to be self-dual iff there exists an inner product—that is, a positive definite bilinear form $\langle . , . \rangle: X \times X \rightarrow \mathbb{R}$—such that $K = K^*_{\langle . , . \rangle}$. Here $K^*_{\langle . , . \rangle}$ is the dual with respect to the inner product $\langle . , . \rangle$, that is $K^*_{\langle . , . \rangle} := \{ y \in X: \forall x \in X ~\langle y, x \rangle > 0 \}$. So in asking for a proof that a domain of positivity is a self-dual cone, we are asking whether some inner product $\langle . , . \rangle$ with respect to which $K$ is self-dual exists. Above, I considered the case $K=Y$, and called the inner product I was looking for, $B'$.

Does anyone know, or can anyone come up with, a proof?

Answer 1

I believe that the statement you want is not true. In $X=\mathbb R^3$, begin with the standard cone $x^2+y^2<z^2$ and perturb it so that the resulting cone $K$ is symmetric to its Euclidean dual through the $yz$-plane and has no affine symmetries (that is, no nontrivial linear maps that map it to itself). As your argument shows, this cone is self-dual w.r.t. $-x^2+y^2+z^2$.

I claim that this is a unique non-degenerate form which makes $K$ self-dual. Indeed, the dual cone is naturally (canonically) defined in the dual space $X^*$. A bilinear form defines a linear isomorphism between $X^*$ and $X$, and the dual cone in $X$ is the image of the canonical dual cone under this isomorphism. Since $K$ has no affine symmetries, there is only one linear map from $X^*$ to $X$ that sends the canonical dual cone to $K$. Therefore there is only one non-degenerate bilinear form that makes $K$ self-dual. And it is not positive.

Answer 2

In the book by Faraut and Koranyi "Analysis on symmetric cones" (Exercise 10 on p. 21 and note on p. 23) it is stated that it was Vinberg in

• Vinberg, Homogeneous cones, Soviet Math. Dokl. 1, (1960) 787-790

who discovered that there are homogeneous cones which are not self dual with respect to any scalar product.

Answer 3

Here's what's true instead of the claim that domains of positivity are self-dual cones.

$\mathbf{Proposition:}$ $Y$ is a domain of positivity for a nondegenerate symmetric bilinear form $B$ if and only if it is an open cone whose dual, according to the Euclidean inner product $E$ associated with a basis orthonormalizing the form, is its image under reflection of $X_-$ through $X_+$, the ``negative and positive eigenspaces'' associated with the form in this basis.

$\mathbf{Proof:}$ We'll write $v,v'$ for vectors in $X$. We'll use an orthonormal basis as described above, in which the form is diagonal with diagonal elements $\pm 1$, writing $v = (x,t)$ for a decomposition with $x$ in the span (call it $X_+$) of the basis vectors with $B(e_i, e_i) = 1$, and $t$ in the span (call it $X_-$) of the basis vectors with $B(e_i, e_i) = -1$. Let $S$ be the linear map $(x,t) \mapsto (x, -t)$, i.e. reflection of the subspace $X_-$ through the subspace $X_+$. Note that $E(x,y) := B(x,Sy)$ is a positive semidefinite symmetric nondegenerate bilinear form.
Also, note that for all $v,v'$, $B(Sv, Sv') = B(v,v')$, i.e. the form $B$ is reflection-symmetric.

For "if": the definition of $Y^\ast$ says it is maximal such that $E(Y^\ast,Y) > 0$. But since $Y=SY$, it is also maximal such that $E(SY,Y) \equiv B(Y,Y) > 0$, i.e., it is a domain of positivity of $B$.

For ``only if'': let $Y$ be a domain of positivity for $B$. For every $y$ in the boundary $\partial Y$ of $Y$, the hyperplane $H_y := \{x: B(x,y) = 0\}$ is a supporting hyperplane for the cone $Y$, and these are all the supporting hyperplanes. But it's standard convex geometry that the supporting hyperplanes of a proper convex cone $Y$ are the precisely the zero-sets of the linear functionals that constitute the boundary of $Y$'s dual cone. We have $H_y = \{x: B(x,y) \equiv E(x,Sy) = 0\}$; that is, this hyperplane is just the plane normal to $Sy$ according to the Euclidean inner product. That is to say, the vectors $Sy$, for $y \in \partial Y$ generate the closure of the cone $Y^\ast$ dual to $Y$ according to the Euclidean inner product $E$. I.e., $Y^\ast = SY$. $\diamond$

Offline (or rather, off-math-overflow) correspondence with Will Jagy helped stimulate this solution. He gave another example---which I'd come up with a few weeks ago, but forgotten about---of a DOP for $xx' + yy' - zz'$---namely, the positive orthant generated by $(0, 1, 0)$, $(1, 0, 1)$ and $(1, 0, -1)$ (or in his dual description, defined by inequalities $x > z$, $x > -z$, $y > 0$), which is of course not isomorphic to an ice-cream cone, but is symmetric under reflection through the xy plane. The hypothesis that the DOPs were precisely the self-dual cones symmetric under reflection suggested itself to me, and attempts to prove the hypothesis ended up providing the proof of the proposition above.