---
title: Weakly Convex Sets in Geometry
url: https://www.emergentmind.com/topics/weakly-convex-sets
type: topic
---

# Weakly Convex Sets in Geometry

Searching arXiv for recent and foundational papers on weakly convex sets and related notions.
Weakly convex sets are a family of generalized convexity notions that arise in several mathematically distinct settings. In the literature represented here, the term denotes at least four non-equivalent constructions: $\theta$-convex subsets of complete metric spaces, weakly $m$-convex subsets of Euclidean space defined by escaping affine planes, weakly convex sets in Banach spaces quantified by a modulus of nonconvexity, and weakly convex vertex sets in graphs defined by the existence of internal geodesics. What these notions share is a relaxation of classical convexity that preserves some interval, midpoint, projection, or geodesic structure while allowing disconnectedness or controlled nonconvexity [2105.06251].

## 1. Terminological scope and principal definitions

The phrase *weakly convex* is not uniform across the literature. The following table summarizes the principal definitions appearing in recent arXiv work.

| Setting | Definition | Characteristic feature |
|---|---|---|
| Complete metric space $(X,D)$ | $C\subseteq X$ is $\theta$-convex if $C$ is closed and, whenever $x,y\in C$ satisfy $D(x,y)\le \theta$, then $I(x,y)\subseteq C$ | Local geodesic closure up to scale $\theta$ |
| Euclidean space $\mathbb R^n$ | An open set is weakly $m$-convex if for every boundary point there exists an $m$-plane through that point disjoint from the set; a closed set is defined by approximation from the outside | Boundary-separation by affine planes |
| Banach space $(E,\|\cdot\|)$ | A closed set $A$ is weakly convex with modulus $\gamma_A$ if for every $\epsilon\in[0,d)$ one has $0\le \gamma_A(\epsilon)<\epsilon/2$ | Quantified midpoint filling |
| Connected graph $G=(V,E)$ | $C\subseteq V$ is weakly convex if for every $u,v\in C$ there exists a $u$-$v$ geodesic whose vertices lie in $C$ | Existence of one internal shortest path |

In the metric-space framework of "Learning Weakly Convex Sets in Metric Spaces" [2105.06251], the metric interval is
\[
I(x,y)=\{\,z\in X : D(x,z)+D(z,y)=D(x,y)\,\},
\]
and weak convexity is parameterized by a threshold $\theta\ge 0$. In the Euclidean generalized-convexity framework of Osipchuk and earlier work by Dakhil–Zelinskii–Klishchuk, the relevant objects are $m$-dimensional affine planes through boundary points [2111.01574], [1703.06785]. In Banach spaces, Balashov–Repovš replace exact midpoint closure by a scale-dependent defect $\gamma_A$ [1007.0162]. In graph theory, weak convexity is strictly weaker than geodesic convexity because only one shortest path must remain inside the set [2501.15524].

A common misconception is that these are merely reformulations of the same idea. The cited literature instead treats them as distinct generalized convexities adapted to different ambient geometries.

## 2. Metric weak convexity in complete metric spaces

Let $(X,D)$ be a complete metric space and $\theta\ge 0$. A subset $C\subseteq X$ is called $\theta$-convex, or weakly convex, if it is closed and if every pair $x,y\in C$ with $D(x,y)\le \theta$ contains its full metric interval:
\[
I(x,y)\subseteq C.
\]
This produces a scale-indexed family $\mathcal C_\theta$ of weakly convex sets. At $\theta=0$, one has $\mathcal C_0=2^X$, while as $\theta$ increases the class shrinks monotonically, and in the limit $\mathcal C_\infty$ coincides with ordinary geodesic convexity [2105.06251].

The family $\mathcal C_\theta$ is closed under arbitrary intersections and therefore forms a closure system. Its associated closure operator,
\[
\operatorname{wconv}_{\theta}(A)=\bigcap\{\,C\in\mathcal C_{\theta}:A\subseteq C\},
\]
maps any $A\subseteq X$ to the smallest $\theta$-convex superset. This weakly convex hull admits a canonical decomposition into well-separated blocks. Two points are $\theta$-connected if they can be linked by a finite chain of pairwise distances at most $\theta$. A $\theta$-convex set $C$ then decomposes uniquely as
\[
C=B_1\cup\cdots\cup B_m,
\]
where each block is nonempty, $\theta$-connected, $\theta$-convex, and distinct blocks are at distance $>\theta$ [2105.06251].

This decomposition is structurally important because block decompositions only merge as $\theta$ increases; they never split. Under a further ambient assumption called *blockwise convexity*, each $\theta$-connected $\theta$-convex set is globally convex. The paper gives four representative examples: the Hamming cube $\{0,1\}^n$ for $\theta\ge 2$, where blocks are Boolean subcubes; the unit cube $[0,1]^d$ under $L_1$, where blocks are axis-aligned hyperrectangles; the Euclidean plane $\mathbb R^2$ under $L_2$, where blocks are ordinary convex polygons; and arbitrary finite graphs under graph distance, where blocks are geodesically convex subgraphs [2105.06251].

This framework is noteworthy because it permits several disconnected regions while retaining an exact interval structure below the threshold $\theta$. A plausible implication is that the parameter $\theta$ acts as a geometric regularizer interpolating between arbitrary subsets and full convexity.

## 3. Algorithmic learning and weakly convex hulls

The principal algorithmic use of metric weak convexity in [2105.06251] is the consistent hypothesis finding problem. Given labeled examples $(E^+,E^-)$, the goal is to find the largest threshold $\theta$ for which the weakly convex hull $\operatorname{wconv}_\theta(E^+)$ contains all positive examples and avoids all negative ones. The paper provides a domain-independent intensional algorithm based on successive block merging.

The representation scheme assumes that for each finite $A\subseteq X$ and $\theta$, one can enumerate the block family $\mathcal B_\theta(A)$ in polynomial time and encode each block by a concise representation $\rho_\theta(B)$. Three primitives are required: `Distance`, which returns the infimum distance between represented blocks; `Join`, which merges two blocks at distance at most $\theta$ into the block encoding of $\operatorname{wconv}_\theta(B_1\cup B_2)$; and `Membership`, which tests whether a point belongs to the represented set. Under mild blockwise convexity assumptions, all of these primitives run in polynomial time [2105.06251].

Correctness follows from monotonicity in $\theta$ together with the fact that the successive merges reproduce the block decomposition of $\operatorname{wconv}_{\theta_i}(E^+)$. If the primitive costs are $T_D,T_J,T_M$ and $|E^+|=m_\oplus$, $|E^-|=m_\ominus$, the overall complexity is
\[
O\bigl(m_\oplus^2\log m_\oplus\;+\;m_\oplus\,T_S\;+\;m_\oplus^2\,T_D\;+\;m_\oplus\,T_J\;+\;m_\oplus\,m_\ominus\,T_M\bigr),
\]
where $T_S$ is the cost of creating singleton blocks [2105.06251].

The paper works out several nontrivial examples. In the Hamming cube, each block is a Boolean subcube encoded by a conjunction, and the running time is
\[
O(n\,m_\oplus(m_\oplus+m_\ominus)).
\]
In $([0,1]^d,D_1)$, blocks are axis-aligned hyperrectangles encoded by minimum and maximum points, with total time
\[
O(m_\oplus^2\log m_\oplus + m_\oplus^2 d + m_\oplus m_\ominus d).
\]
In $(\mathbb R^2,D_2)$, each finite $\theta$-connected hull is a convex polygon with vertices from the input, yielding total time
\[
O(m_\oplus^2\log m_\oplus + m_\oplus m_\ominus\log m_\oplus).
\]
The paper emphasizes that without the weak convexity constraint the corresponding problems are computationally intractable [2105.06251].

An extensional variant is also given for finite graphs with geodesic distance. After precomputing all pair distances, it iteratively adds intervals $I(x,y)$ whenever $0<D_g(x,y)\le \theta_i$. The total complexity is $O(T_P(G)+n^3)$ time and $O(n^2)$ space, where $T_P(G)$ is the preprocessing cost. Empirical tests on grid graphs, Delaunay triangulations, and random graphs are reported to achieve at least $0.9$ accuracy in vertex classification tasks with $20$–$40$ labeled vertices [2105.06251].

## 4. Weakly \(m\)-convex sets in Euclidean space

In the Euclidean generalized-convexity literature, weakly $m$-convex sets are defined by escaping affine planes rather than metric intervals. An open set $G\subset \mathbb R^n$ is weakly $m$-convex if for every boundary point $x\in\partial G$ there exists an $m$-dimensional plane $L$ with $x\in L$ and $L\cap G=\varnothing$. A closed set $E\subset\mathbb R^n$ is weakly $m$-convex if it is approximated from the outside by a nested family of open weakly $m$-convex sets whose intersection is $E$ [2111.01574]. A point $x\in\mathbb R^n\setminus E$ is an $m$-nonconvexity point if every $m$-plane through $x$ intersects $E$; the set of all such points is denoted $(E)_m^\triangle$ [2111.01574].

This definition yields strong topological constraints. Osipchuk proves that any closed, weakly $(n-1)$-convex set in $\mathbb R^n$ with non-empty set of $(n-1)$-nonconvexity points consists of not less than three connected components [2111.01574]. In the planar case, if $E\subset\mathbb R^2$ is closed, weakly $1$-convex, has finitely many components, and $\operatorname{Int}E\neq\varnothing$, then $\operatorname{Int}E$ is itself weakly $1$-convex [2111.01574]. A later result states more generally that if a closed weakly $1$-convex set in $\mathbb R^n$ has non-empty interior, then the interior is weakly $1$-convex [2412.01022].

The same literature establishes existence in the opposite direction: for each $n\ge 3$ and each $1\le m\le n-2$, there exist open domains and closed connected sets in $\mathbb R^n$ that are weakly $m$-convex but not $m$-convex [2111.01574]. Thus the component lower bound for the extreme case $m=n-1$ does not extend unchanged to lower-dimensional escaping planes.

A representative planar construction is the “triangle + three trapezoids” example. Starting from an open equilateral triangle and placing trapezoids in the three angular regions, one obtains open sets $E_t$ that are weakly $1$-convex but not $1$-convex, with the inner triangle $a_tb_tc_t$ serving as the set of $1$-nonconvexity points. The closure $\overline{E_0}$ is then a closed, connected weakly $1$-convex set in $\mathbb R^2$ that is not $1$-convex [2111.01574].

A further structural theorem for open weakly $1$-convex sets states that if the set of $1$-nonconvexity points is non-empty, then that set is itself open; the same statement holds for weakly $1$-semiconvex sets and their nonsemiconvexity regions [2412.01022]. This makes the nonconvexity locus a geometric object in its own right rather than merely an exceptional set.

## 5. Banach-space weak convexity and the modulus of nonconvexity

In Banach spaces, weak convexity is formulated quantitatively. For a closed set $A\subset E$ and $\epsilon\in[0,d)$, the modulus of nonconvexity is
\[
\gamma_A(\epsilon)
=
\inf\Bigl\{\gamma\ge 0\ \Big|\
B_\gamma\!\Bigl(\tfrac{x_1+x_2}{2}\Bigr)\cap A\neq\emptyset
\ \ \forall x_1,x_2\in A,\ \|x_1-x_2\|\le \epsilon
\Bigr\},
\qquad \gamma_A(0)=0.
\]
The set $A$ is weakly convex with modulus $\gamma_A$ on $[0,d)$ if for every $\epsilon\in[0,d)$ one has
\[
0\le \gamma_A(\epsilon)<\frac{\epsilon}{2}.
\]
Equivalently, whenever $\|x_1-x_2\|=\epsilon<2d$, there exists $z\in A$ such that $\|z-\tfrac{x_1+x_2}{2}\|\le \gamma_A(\epsilon)$ [1007.0162].

This notion generalizes Vial’s weak convexity in Hilbert spaces and is closely related to Clarke’s proximal smoothness. In a Hilbert space $H$, if $\gamma_A(\epsilon)=R\,\delta_H(\epsilon/R)$ with
\[
\delta_H(t)=1-\sqrt{1-t^2/4},
\]
then the new definition is equivalent to Vial’s. In uniformly smooth Banach spaces, proximal smoothness at radius $d$ implies $\gamma_A(\epsilon)<\epsilon/2$ for all $\epsilon<2d$, and under the stronger inequality
\[
d\,\delta_E(\epsilon/d)>\gamma_A(\epsilon)
\]
one recovers uniqueness of nearest-point projections and hence proximal smoothness [1007.0162]. Classical convexity is the degenerate case $\gamma_A(\epsilon)\equiv 0$.

Several analytic consequences follow. If $A$ is weakly convex with modulus $\gamma_A$ and
\[
d\,\delta_E(\epsilon/d)>\gamma_A(\epsilon)\qquad
\forall \epsilon\in (0,\min\{2d,\operatorname{diam}A\}),
\]
then every point of the $d$-neighborhood $U_d(A)$ has a unique nearest-point projection $P_A(x)$ [1007.0162]. Under an additional technical condition relating $\delta_E$ and $\gamma_A$, the projection is uniformly continuous, with a Hölder-type modulus $t_E(s)\asymp \sqrt{s}$ as $s\to 0$ [1007.0162].

The same midpoint-control mechanism yields a continuous retraction theorem, Hausdorff continuity of intersections of set-valued maps when one family is uniformly weakly convex and the other uniformly convex, and an affirmative splitting result for uniformly continuous selections [1007.0162]. In this setting, weak convexity is therefore less a combinatorial relaxation than a quantitative geometric regularity condition.

## 6. Graph-theoretic weak convexity and related notions

For a connected graph $G=(V,E)$ with geodesic distance $d_G$, a subset $C\subseteq V$ is weakly convex if for every $u,v\in C$ there exists a $u$-$v$ geodesic $P$ such that all vertices of $P$ lie in $C$ [2501.15524]. This is weaker than geodesic convexity, which requires containment of all shortest paths, and it is used in the definition of the outer-weakly convex domination number $\widetilde\gamma_{wcon}(G)$, the minimum size of a dominating set $D$ such that $V\setminus D$ is weakly convex [2501.15524].

The 2025 study of graph products derives bounds for $\widetilde\gamma_{wcon}$ under the Cartesian, strong, and lexicographic products. For example, if $|V(G)|=m$ and $|V(H)|=n$, then
\[
\min\{m,n\}
<
\widetilde\gamma_{wcon}(G\square H)
<
\min\bigl\{\widetilde\gamma_{wcon}(G)\,n,\ \widetilde\gamma_{wcon}(H)\,m\bigr\},
\]
and for the lexicographic product one has
\[
\widetilde\gamma_{wcon}(G)
\le
\widetilde\gamma_{wcon}(G\circ H)
\le
\widetilde\gamma_{wcon}(G)+P_G,
\]
where $P_G$ is defined from isolated vertices in a minimum outer $c$-dominating set [2501.15524]. These results show that weak convexity retains enough path structure to support product-graph domination theory.

The Euclidean weakly $m$-convex literature also connects to “shadow” problems. Dakhil–Zelinskii–Klishchuk study the role of weakly $m$-convexity in the geometry of shadows cast by balls and prove, among other results, a lower bound
\[
N_{\min}(\mathbb R^3,S^2)\ge 15
\]
for the number of pairwise disjoint equal-radius balls needed to cast a tangent shadow at every point of the sphere [1703.06785]. Although this problem is not a definition of weak convexity, it illustrates how weakly convex structures interact with visibility and avoidance phenomena.

A related but distinct notion is weak semiconvexity. In the plane, weakly $1$-semiconvex sets replace escaping lines by escaping rays. Open weakly $1$-semiconvex but non-$1$-semiconvex sets with smooth boundary require at least four connected components, while closed weakly $1$-semiconvex but non-$1$-semiconvex sets require at least three connected components; these bounds are sharp in the cited constructions [2002.03422], [1711.04757]. This comparison is useful because weak $1$-convexity and weak $1$-semiconvexity are adjacent but not identical generalized convexities.

Across these settings, weakly convex sets function as controlled relaxations of convexity adapted to different geometric primitives: intervals in metric spaces, affine planes in Euclidean space, midpoint defects in Banach spaces, and internal geodesics in graphs. The diversity of definitions is not accidental; it reflects the fact that “convexity” itself depends on which ambient notion of straightness or separation is treated as fundamental.

Source: https://www.emergentmind.com/topics/weakly-convex-sets