Slice Cone: Geometry, Algebra & Function Theory

• Slice cones are unifying constructs that represent unions or slices of higher-dimensional cones, key in convex analysis and function theory.
• They facilitate cone lifts and factorization certificates in conic optimization, transforming complex problems into tractable semidefinite programs.
• In algebraic settings, slice cones define natural domains for slice-regular functions, supporting analytic continuation in non-commutative and non-associative algebras.

A slice cone is a unifying geometric and algebraic construct that appears across convex analysis, real associative and non-associative algebra, and the theory of slice regular functions in hypercomplex settings. In its various incarnations, the slice cone formalism encodes a geometric or function-theoretic object as a union, projection, or slice of higher-dimensional structures—typically cones, affine slices of cones, or unions of “slices” indexed by a family of complex structures or imaginary units. Slice cones play a central role in cone lifts of convex sets, in the structure theory of function spaces over non-commutative or non-associative algebras, and in the algebraic analysis of conic optimization.

1. Slice Cones in Convex Geometry and Cone Lifts

The slice cone viewpoint in convex geometry arises from the observation that every closed convex set $C\subset\mathbb R^n$ admits a representation as the projection of an affine slice of a higher-dimensional cone $K\subset\mathbb R^m$, provided both $C$ and its recession cone $0^+C$ are realized as slices of the same $K$. More specifically, for an affine subspace $L\subset\mathbb R^m$ and a linear projection $\pi:\mathbb R^m\to\mathbb R^n$, $C$ is a $K$-lift if $C=\pi(K\cap L)$ and $0^+C = \pi(K\cap 0^+L)$. This generalizes classical extended formulations and underlies tight connections between geometric representations and algebraic slack operator factorizations (Wang et al., 2014, Gouveia et al., 2011).

The notion of a "slice" appears here as an affine section of a cone, and the essential insight is that certain algebraic certificates (in terms of cone factorizations of slack operators) are necessary and sufficient for such lifted representations. The slice cone thus provides a powerful mechanism for understanding the expressiveness and computational tractability of linear and semidefinite relaxations of optimization problems.

2. Projective and Affine Slices in Conic Optimization

In conic linear programming, the concept of a slice cone becomes explicit in the duality structure of cones of linear operators acting on symmetric cones, such as the cone of $Z$-transformations on the Lorentz cone $\mathcal L^n$. The dual cone $Z(\mathcal L^n)^*$ can be represented both as a slice of the semidefinite cone and as a slice of a completely positive cone:

$$Z(\mathcal{L}^n)^* = \{ X\in S^n_+ : \langle J, X\rangle=0\} = \{ X\in K_{\partial\mathcal{L}^n}: \langle J, X\rangle=0\}$$

where $J=\mathrm{diag}(1,-1,\dots,-1)$ and $K_{\partial\mathcal{L}^n}$ is the completely positive cone induced by the boundary of the Lorentz cone. This slice—hyperplane section—of the semidefinite cone enables the reduction of conic linear programming over completely positive cones to standard semidefinite programming with a single additional linear constraint, providing a method to "lift" a generally hard optimization problem into a tractable one (Németh et al., 2019).

3. Slice Cones in Non-Associative and Associative Algebras

In real associative algebras $A$ (e.g., quaternions, Clifford algebras) and real alternative division algebras (e.g., octonions), the "quadratic cone" or slice cone is defined by considering the set $S$ of square roots of $-1$ in $A$. Given a suitable inner product, the subset $S_0$ of orthogonal complex structures gives rise to the slice cone:

$$Q := \bigcup_{s\in S_0} C_s, \quad C_s := \{\alpha + \beta s: \alpha,\beta\in\mathbb R\}$$

This union of real 2-planes indexed by $S_0$ is endowed with a natural complex structure and serves as the canonical domain for slice-regular function theory. In Clifford algebras, for instance, $Q$ has real dimension $2 + N/2$ (with $N$ the dimension of $A$), is diffeomorphic to $S_0\times\mathbb R^2$, and is a complex manifold locally modeled on $\mathbb C\times S_0$ (Mongodi, 2019).

4. Slice Cones and Function Theory: Slice-regularity

The geometric structure of slice cones underlies the modern theory of slice-regular functions for hypercomplex variables. In octonionic analysis, the $n$-dimensional quadratic cone (or slice cone) is

$$\mathcal Q_n := \bigcup_{I\in S} C_I^n = \{ x + y I : x,y\in\mathbb R^n, I\in S, I^2 = -1\}$$

where each $C_I^n$ is a "slice" isomorphic to $\mathbb C^n$. Axially symmetric domains in $\mathcal Q_n$ (unions of all $x + y J$ for fixed $x,y$ and varying $J \in S$) provide the right topological and algebraic setting for representation formulas, identity principles, and power series for slice-regular and weak slice-regular functions (Dou et al., 2020, Dou et al., 2020).

These slice-cone domains permit consistent holomorphic extension procedures along slices and encode the necessary symmetry to transfer complex analysis phenomena into non-commutative or non-associative settings. Slice-topologies are defined so that slice-open sets restrict to open subsets in each $C_I^n$, and the union structure is critical for algebraic properties of regularity.

5. Generalized Slice-cone Construction in Real Linear Settings

A broad generalization of the slice-cone concept applies to arbitrary real vector spaces with even dimension. For a subset $\mathcal C$ of complex structures $T$ (i.e., $T^2 = -\mathrm{Id}$), the $d$-dimensional weak slice-cone is

$$\mathcal{W}_\mathcal{C}^d := \bigcup_{I\in\mathcal C} \mathbb C_I^d \subset (\mathbb R^{2n})^d$$

where each $\mathbb C_I^d$ is modeled on $(x_1 + y_1 I, \dots, x_d + y_d I)$ with $x_j, y_j \in \mathbb R$. The slice-topology $\tau_s$ defines open sets via openness in the Euclidean topology of each slice. This framework encompasses classical settings (quaternions, octonions, Clifford algebras, sedenions) and permits the systematic study of weak slice-regular functions and their representation formulae in several variables, via the Moore–Penrose inverse matrix formalism (Dou et al., 2020).

This generality enables the transfer of the slice-function theory to linear algebraic contexts, avoiding issues of non-associativity, and facilitates results such as multivariable representation formulas and identity principles that hold in all LSCS algebras (left slice complex structure algebras).

6. Slice Cone Algorithms and Explicit Constructions

The classical projective geometry of conics as plane sections of circular cones also fits the slice-cone paradigm. Any nondegenerate conic $f(x) = x^\mathrm{T} A x + b^\mathrm{T}x + c = 0$ can be realized as the intersection of the plane $z=0$ with a circular cone in $\mathbb R^3$ whose apex and axis are given by explicit formulas involving orthogonal diagonalization, confocal parameter computations, and matrix algebra:

$(X - p)^\mathrm{T} Q (X - p) = 0$

where $Q = r r^\mathrm{T} - \cos^2\theta\, I_3$, $p$ is the apex, $r$ the direction vector, and the half-angle $\theta$ is determined via the confocal quadric parameterization. The “slice-cone” is the geometric locus sliced by the plane, precisely recovering the given conic section (Armstrong, 2017).

7. Significance and Unified Perspective

The slice-cone concept synthesizes geometric, algebraic, and functional viewpoints. In convex optimization, it unifies cone-lift and factorization theories; in real algebra, it models domains for slice-regularity; in geometry, it underpins analytic representations of classical objects. The slice cone viewpoint highlights the essential structure in extended formulations and function theories, providing both constructive algorithms and deep theoretical insight into the nature of higher-dimensional projections and holomorphicity in non-classical settings (Wang et al., 2014, Gouveia et al., 2011, Dou et al., 2020, Dou et al., 2020, Mongodi, 2019, Armstrong, 2017).