Capra-Convex Sets: Ray-Normalized Convexity

• Capra-convex sets are ray-invariant subsets of ℝⁿ defined via directional normalization, yielding a convex trace on the unit sphere-plus-origin.
• They generalize classical convex cones and reveal hidden convexity in problems such as sparsity, support constraints, and low-rank optimization.
• Their structure relies on the chosen source norm, allowing nonconvex problems to be reformulated as convex ones over normalized directions.

Capra-convex sets are a class of sets in $\mathbb{R}^n$ defined through a generalized Fenchel–Moreau theory in which the usual scalar product is replaced by a coupling that is constant along primal rays. Relative to a chosen source norm, the geometry is organized by directions rather than amplitudes: a set is determined by its normalized trace on the unit sphere-plus-origin, and Capra-convexity is the stability of its indicator under the associated biconjugacy. This framework was developed to capture the hidden convexity of ray-invariant objects such as the $\ell_0$ pseudonorm and its cardinality sublevel sets, and it now provides a set-level theory in which closed convexity appears after normalization rather than in the ambient Euclidean space (Franc et al., 8 Sep 2025).

1. Coupling-based definition

Fix a norm $\|\cdot\|$ on $\mathbb{R}^n$, called the source norm, with unit sphere and unit ball

$$S=\{x\in\mathbb{R}^n:\|x\|=1\}, \qquad B=\{x\in\mathbb{R}^n:\|x\|\le 1\}.$$

The associated normalization map is

$n:\mathbb{R}^n\to S\cup\{0\}, \qquad n(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0. \end{cases}$

Its decisive property is zero-homogeneity: $$n(\lambda x)=n(x)\qquad \forall \lambda\in\mathbb{R}\setminus\{0\},\ \forall x\in\mathbb{R}^n.$$ The Capra coupling is then

$\mathcal{C}(x,y)=\langle n(x),y\rangle= \begin{cases} \dfrac{\langle x,y\rangle}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0, \end{cases}$

so $\mathcal{C}(\lambda x,y)=\mathcal{C}(x,y)$ for every $\lambda>0$. In this sense, the coupling is constant along primal rays, which is the origin of the name Capra, for “Constant Along Primal Rays” (Franc et al., 8 Sep 2025).

Given $\ell_0$0, its Capra-Fenchel conjugate and biconjugate are

$\ell_0$1

with $\ell_0$2. A function is Capra-convex iff

$\ell_0$3

A set $\ell_0$4 is Capra-convex iff its indicator

$\ell_0$5

is Capra-convex. At the function level, the precursor framework already showed that Capra-convexity is equivalent to factorization through normalization: a function is Capra-convex iff there exists a closed convex $\ell_0$6 such that $\ell_0$7, and for indicators one has $\ell_0$8, so the dual object is governed by the normalized image rather than by the original set (Chancelier et al., 2020).

2. Structural characterization of Capra-convex sets

The set-level characterization is explicit. A set $\ell_0$9 is Capra-convex iff

$\|\cdot\|$0

Equivalently, a Capra-convex set is exactly the inverse image, under normalization $\|\cdot\|$1, of a closed convex subset of $\|\cdot\|$2 determined by its normalized trace. The whole set is therefore reconstructed from its direction set after closed convexification on the sphere-plus-origin (Franc et al., 8 Sep 2025).

For a cone $\|\cdot\|$3, one has

$\|\cdot\|$4

so the fundamental theorem can be read geometrically as follows: normalize the set onto the sphere-plus-origin; the normalized image must be the trace on $\|\cdot\|$5 of a closed convex set. This yields the immediate consequence that every Capra-convex set is a cone, since Capra-convexity forces invariance under positive scaling: $\|\cdot\|$6

The theory also supplies practical necessary conditions. If $\|\cdot\|$7 is Capra-convex, then $\|\cdot\|$8 is a cone, $\|\cdot\|$9 is closed, and

$\mathbb{R}^n$0

When $\mathbb{R}^n$1, this reduces to the efficient test

$\mathbb{R}^n$2

If the source norm ball is rotund, meaning its sphere is exactly the set of extreme points of the ball, then these practical conditions are also sufficient. This holds, for example, for $\mathbb{R}^n$3 with $\mathbb{R}^n$4, including $\mathbb{R}^n$5 (Franc et al., 8 Sep 2025).

3. Ray geometry, norm dependence, and basic examples

Capra-convexity is not ordinary convexity in $\mathbb{R}^n$6. The governing structure is not the Euclidean segment between points, but the ray structure induced by normalization. For arbitrary $\mathbb{R}^n$7,

$\mathbb{R}^n$8

Hence a Capra-convex set is invariant along rays and completely determined by its section on $\mathbb{R}^n$9. This explains why Capra-convex sets are necessarily cones, but need not be convex in the usual sense and need not even be closed (Franc et al., 8 Sep 2025).

The dependence on the source norm is substantial. The same cone may be Capra-convex for one norm and not for another. In $$S=\{x\in\mathbb{R}^n:\|x\|=1\}, \qquad B=\{x\in\mathbb{R}^n:\|x\|\le 1\}.$$0, define

$$S=\{x\in\mathbb{R}^n:\|x\|=1\}, \qquad B=\{x\in\mathbb{R}^n:\|x\|\le 1\}.$$1

$$S=\{x\in\mathbb{R}^n:\|x\|=1\}, \qquad B=\{x\in\mathbb{R}^n:\|x\|\le 1\}.$$2

$$S=\{x\in\mathbb{R}^n:\|x\|=1\}, \qquad B=\{x\in\mathbb{R}^n:\|x\|\le 1\}.$$3

With source norm $$S=\{x\in\mathbb{R}^n:\|x\|=1\}, \qquad B=\{x\in\mathbb{R}^n:\|x\|\le 1\}.$$4, $$S=\{x\in\mathbb{R}^n:\|x\|=1\}, \qquad B=\{x\in\mathbb{R}^n:\|x\|\le 1\}.$$5 is not Capra-convex because $$S=\{x\in\mathbb{R}^n:\|x\|=1\}, \qquad B=\{x\in\mathbb{R}^n:\|x\|\le 1\}.$$6, $$S=\{x\in\mathbb{R}^n:\|x\|=1\}, \qquad B=\{x\in\mathbb{R}^n:\|x\|\le 1\}.$$7 is Capra-convex even though it is not convex in the ordinary sense, and $$S=\{x\in\mathbb{R}^n:\|x\|=1\}, \qquad B=\{x\in\mathbb{R}^n:\|x\|\le 1\}.$$8 is Capra-convex. With source norm $$S=\{x\in\mathbb{R}^n:\|x\|=1\}, \qquad B=\{x\in\mathbb{R}^n:\|x\|\le 1\}.$$9, $n:\mathbb{R}^n\to S\cup\{0\}, \qquad n(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0. \end{cases}$0 is not Capra-convex, $n:\mathbb{R}^n\to S\cup\{0\}, \qquad n(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0. \end{cases}$1 is also not Capra-convex, and $n:\mathbb{R}^n\to S\cup\{0\}, \qquad n(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0. \end{cases}$2 remains Capra-convex. The divergence of $n:\mathbb{R}^n\to S\cup\{0\}, \qquad n(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0. \end{cases}$3 between $n:\mathbb{R}^n\to S\cup\{0\}, \qquad n(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0. \end{cases}$4 and $n:\mathbb{R}^n\to S\cup\{0\}, \qquad n(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0. \end{cases}$5 is the paper’s clearest illustration that Capra-convexity depends on the source norm (Franc et al., 8 Sep 2025).

A major positive class is provided by classical convex cones. If $n:\mathbb{R}^n\to S\cup\{0\}, \qquad n(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0. \end{cases}$6 is a closed convex cone, then $n:\mathbb{R}^n\to S\cup\{0\}, \qquad n(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0. \end{cases}$7 is Capra-convex. If $n:\mathbb{R}^n\to S\cup\{0\}, \qquad n(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0. \end{cases}$8 is pointed, then $n:\mathbb{R}^n\to S\cup\{0\}, \qquad n(x)= \begin{cases} \dfrac{x}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0. \end{cases}$9 is also Capra-convex. This shows that the theory contains all closed convex cones, but is strictly broader because it also includes cones that are not convex in the ordinary sense (Franc et al., 8 Sep 2025).

4. Hidden convexity of sparsity and support-constrained sets

The motivation for Capra-convexity comes from sparsity. The $$n(\lambda x)=n(x)\qquad \forall \lambda\in\mathbb{R}\setminus\{0\},\ \forall x\in\mathbb{R}^n.$$0 pseudonorm

$$n(\lambda x)=n(x)\qquad \forall \lambda\in\mathbb{R}\setminus\{0\},\ \forall x\in\mathbb{R}^n.$$1

is constant along nonzero rays, so ordinary Fenchel conjugacy is not adapted to it. The Euclidean precursor introduced the $$n(\lambda x)=n(x)\qquad \forall \lambda\in\mathbb{R}\setminus\{0\},\ \forall x\in\mathbb{R}^n.$$2-Capra coupling and proved the exact identities

$$n(\lambda x)=n(x)\qquad \forall \lambda\in\mathbb{R}\setminus\{0\},\ \forall x\in\mathbb{R}^n.$$3

so the lower sparsity level sets and the $$n(\lambda x)=n(x)\qquad \forall \lambda\in\mathbb{R}\setminus\{0\},\ \forall x\in\mathbb{R}^n.$$4 pseudonorm itself are convex in the generalized Capra sense (Chancelier et al., 2019).

The later norm-general formulation established that, when both the source norm and its dual norm are orthant-strictly monotonic, every nondecreasing finite-valued function of the support mapping is Capra-convex. In particular, for suitable source norms one has

$$n(\lambda x)=n(x)\qquad \forall \lambda\in\mathbb{R}\setminus\{0\},\ \forall x\in\mathbb{R}^n.$$5

and more generally every such function can be written as the composition of a proper convex lower semicontinuous function on $$n(\lambda x)=n(x)\qquad \forall \lambda\in\mathbb{R}\setminus\{0\},\ \forall x\in\mathbb{R}^n.$$6 with the normalization map. This is the hidden-convexity interpretation of support-dependent combinatorial penalties (Chancelier et al., 2020).

The subdifferential theory makes the norm dependence sharper. For source norm $$n(\lambda x)=n(x)\qquad \forall \lambda\in\mathbb{R}\setminus\{0\},\ \forall x\in\mathbb{R}^n.$$7, $$n(\lambda x)=n(x)\qquad \forall \lambda\in\mathbb{R}\setminus\{0\},\ \forall x\in\mathbb{R}^n.$$8, the $$n(\lambda x)=n(x)\qquad \forall \lambda\in\mathbb{R}\setminus\{0\},\ \forall x\in\mathbb{R}^n.$$9 pseudonorm is everywhere Capra-subdifferentiable and Capra-convex. For $\mathcal{C}(x,y)=\langle n(x),y\rangle= \begin{cases} \dfrac{\langle x,y\rangle}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0, \end{cases}$0, it is not Capra-convex, and its Capra-biconjugate is $\mathcal{C}(x,y)=\langle n(x),y\rangle= \begin{cases} \dfrac{\langle x,y\rangle}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0, \end{cases}$1 at $\mathcal{C}(x,y)=\langle n(x),y\rangle= \begin{cases} \dfrac{\langle x,y\rangle}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0, \end{cases}$2 and $\mathcal{C}(x,y)=\langle n(x),y\rangle= \begin{cases} \dfrac{\langle x,y\rangle}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0, \end{cases}$3 elsewhere. For $\mathcal{C}(x,y)=\langle n(x),y\rangle= \begin{cases} \dfrac{\langle x,y\rangle}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0, \end{cases}$4, it is also not Capra-convex, and its biconjugate is

$\mathcal{C}(x,y)=\langle n(x),y\rangle= \begin{cases} \dfrac{\langle x,y\rangle}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0, \end{cases}$5

Thus the endpoint norms fall outside the exact biconjugacy regime that holds for $\mathcal{C}(x,y)=\langle n(x),y\rangle= \begin{cases} \dfrac{\langle x,y\rangle}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0, \end{cases}$6 (Franc et al., 2021).

At the set level, the current theory incorporates these sparse objects directly: if $\mathcal{C}(x,y)=\langle n(x),y\rangle= \begin{cases} \dfrac{\langle x,y\rangle}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0, \end{cases}$7 is Capra-convex, then every sublevel set

$\mathcal{C}(x,y)=\langle n(x),y\rangle= \begin{cases} \dfrac{\langle x,y\rangle}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0, \end{cases}$8

is Capra-convex. Since $\mathcal{C}(x,y)=\langle n(x),y\rangle= \begin{cases} \dfrac{\langle x,y\rangle}{\|x\|}, & x\neq 0,\[1mm] 0, & x=0, \end{cases}$9 is Capra-convex for suitable source norms, the cardinality-constrained sets

$\mathcal{C}(\lambda x,y)=\mathcal{C}(x,y)$0

are Capra-convex. This is one of the principal reasons the framework is considered relevant to statistics, machine learning, and sparse optimization (Franc et al., 8 Sep 2025).

5. Conical hulls, spherical convexity, and matrix-space analogues

Because Capra-convexity is ray-based, conical hulls play a central role. If $\mathcal{C}(\lambda x,y)=\mathcal{C}(x,y)$1 is Capra-convex, then

$\mathcal{C}(\lambda x,y)=\mathcal{C}(x,y)$2

hence minimization over a set $\mathcal{C}(\lambda x,y)=\mathcal{C}(x,y)$3 can be rewritten over $\mathcal{C}(\lambda x,y)=\mathcal{C}(x,y)$4. The set-theoretic counterpart is that, if $\mathcal{C}(\lambda x,y)=\mathcal{C}(x,y)$5 is compact with

$\mathcal{C}(\lambda x,y)=\mathcal{C}(x,y)$6

and either the source norm ball is rotund or $\mathcal{C}(\lambda x,y)=\mathcal{C}(x,y)$7 is convex, then

$\mathcal{C}(\lambda x,y)=\mathcal{C}(x,y)$8

is Capra-convex. The assumptions matter: if $\mathcal{C}(\lambda x,y)=\mathcal{C}(x,y)$9, or if $\lambda>0$0 is not compact, the conical hull may fail to be Capra-convex (Franc et al., 8 Sep 2025).

The relation to spherical convexity is close but not identical. A subset $\lambda>0$1 is spherically-convex if

$\lambda>0$2

The conical hull of a spherically-convex set is not always Capra-convex, but the closure of that conical hull is. More precisely, if $\lambda>0$3 is spherically-convex, then

$\lambda>0$4

is Capra-convex, and if this cone is pointed, then

$\lambda>0$5

is also Capra-convex. Thus spherical convexity and Capra-convexity are related through cone generation and closure, but the notions are not identical (Franc et al., 8 Sep 2025).

The same ray-normalized convexity has matrix analogues. On $\lambda>0$6, with a source matrix norm, the matrix Capra coupling is

$\lambda>0$7

This produces a generalized convexity theory for rank-based functions. The rank function admits a Capra-biconjugate lower bound in terms of generalized $\lambda>0$8-rank norms, and for the Frobenius norm one has exact equality

$\lambda>0$9

A plausible implication is that Capra-convexity extends naturally from sparse vectors to low-rank matrix models through the same constant-along-rays mechanism (Barbier et al., 2021).

6. Distinctions, limitations, and significance

Several misconceptions are explicitly ruled out by the theory. Capra-convexity is not ordinary convexity in disguise: the cone $\ell_0$00 above is not convex in the ordinary sense but is Capra-convex for the Euclidean source norm. It is not source-norm invariant: the same set may be Capra-convex for $\ell_0$01 and fail for $\ell_0$02. It is not simply spherical convexity, since the conical hull of a spherically-convex set need not be Capra-convex without closure. It is also not epigraph stability in the ordinary sense: if $\ell_0$03 is Capra-convex, then its sublevel sets are Capra-convex, but its epigraph is not generally Capra-convex with respect to the original Capra coupling; instead, epigraph convexity requires a modified coupling on $\ell_0$04 built from $\ell_0$05 (Franc et al., 8 Sep 2025).

The framework’s significance is tied to hidden convexity. If both an objective $\ell_0$06 and the indicator of a constraint set $\ell_0$07 are Capra-convex, then

$\ell_0$08

for some proper lower semicontinuous convex $\ell_0$09. Therefore

$\ell_0$10

so an optimization problem that is nonconvex in the original variables becomes the minimization of a usual convex function over the sphere-plus-origin. The theory is accordingly positioned as relevant to statistics and machine learning, to sparse optimization, and to generalized duality and possible algorithmic frameworks, including cutting-plane methods for Capra-convex functions (Franc et al., 8 Sep 2025).

In this sense, Capra-convex sets are best understood as the set-theoretic layer of a broader generalized convex analysis: they are ray-invariant sets whose normalized directions form the spherical trace of a closed convex set, and they provide the natural feasible regions for optimization problems in which support, sparsity, or rank matter more than amplitude.