Weakly Convex Sets in Geometry

• Weakly convex sets are generalized convex structures that relax classical convexity by preserving key interval, projection, or geodesic properties across different spaces.
• They are characterized by distinct definitions in metric spaces, Euclidean geometry, Banach spaces, and graph theory, enabling controlled nonconvexity tailored to each context.
• These properties lead to practical applications such as efficient weak convex hull computations, domination in graphs, and algorithmic learning with provable complexity bounds.

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 (Stadtländer et al., 2021).

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 $m$0	An open set is weakly $m$1-convex if for every boundary point there exists an $m$2-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 $m$3	A closed set $m$4 is weakly convex with modulus $m$5 if for every $m$6 one has $m$7	Quantified midpoint filling
Connected graph $m$8	$m$9 is weakly convex if for every $(X,D)$0 there exists a $(X,D)$1-$(X,D)$2 geodesic whose vertices lie in $(X,D)$3	Existence of one internal shortest path

In the metric-space framework of "Learning Weakly Convex Sets in Metric Spaces" (Stadtländer et al., 2021), the metric interval is

$(X,D)$4

and weak convexity is parameterized by a threshold $(X,D)$5. In the Euclidean generalized-convexity framework of Osipchuk and earlier work by Dakhil–Zelinskii–Klishchuk, the relevant objects are $(X,D)$6-dimensional affine planes through boundary points (Osipchuk, 2021, Dakhil et al., 2017). In Banach spaces, Balashov–Repovš replace exact midpoint closure by a scale-dependent defect $(X,D)$7 (Balashov et al., 2010). In graph theory, weak convexity is strictly weaker than geodesic convexity because only one shortest path must remain inside the set (Anand et al., 26 Jan 2025).

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)$8 be a complete metric space and $(X,D)$9. A subset $C\subseteq X$0 is called $C\subseteq X$1-convex, or weakly convex, if it is closed and if every pair $C\subseteq X$2 with $C\subseteq X$3 contains its full metric interval: $C\subseteq X$4 This produces a scale-indexed family $C\subseteq X$5 of weakly convex sets. At $C\subseteq X$6, one has $C\subseteq X$7, while as $C\subseteq X$8 increases the class shrinks monotonically, and in the limit $C\subseteq X$9 coincides with ordinary geodesic convexity (Stadtländer et al., 2021).

The family $\theta$0 is closed under arbitrary intersections and therefore forms a closure system. Its associated closure operator,

$\theta$1

maps any $\theta$2 to the smallest $\theta$3-convex superset. This weakly convex hull admits a canonical decomposition into well-separated blocks. Two points are $\theta$4-connected if they can be linked by a finite chain of pairwise distances at most $\theta$5. A $\theta$6-convex set $\theta$7 then decomposes uniquely as

$\theta$8

where each block is nonempty, $\theta$9-connected, $C$0-convex, and distinct blocks are at distance $C$1 (Stadtländer et al., 2021).

This decomposition is structurally important because block decompositions only merge as $C$2 increases; they never split. Under a further ambient assumption called blockwise convexity, each $C$3-connected $C$4-convex set is globally convex. The paper gives four representative examples: the Hamming cube $C$5 for $C$6, where blocks are Boolean subcubes; the unit cube $C$7 under $C$8, where blocks are axis-aligned hyperrectangles; the Euclidean plane $C$9 under $x,y\in C$0, where blocks are ordinary convex polygons; and arbitrary finite graphs under graph distance, where blocks are geodesically convex subgraphs (Stadtländer et al., 2021).

This framework is noteworthy because it permits several disconnected regions while retaining an exact interval structure below the threshold $x,y\in C$1. A plausible implication is that the parameter $x,y\in C$2 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 (Stadtländer et al., 2021) is the consistent hypothesis finding problem. Given labeled examples $x,y\in C$3, the goal is to find the largest threshold $x,y\in C$4 for which the weakly convex hull $x,y\in C$5 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 $x,y\in C$6 and $x,y\in C$7, one can enumerate the block family $x,y\in C$8 in polynomial time and encode each block by a concise representation $x,y\in C$9. Three primitives are required: Distance, which returns the infimum distance between represented blocks; Join, which merges two blocks at distance at most $D(x,y)\le \theta$0 into the block encoding of $D(x,y)\le \theta$1; 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 (Stadtländer et al., 2021).

Correctness follows from monotonicity in $D(x,y)\le \theta$2 together with the fact that the successive merges reproduce the block decomposition of $D(x,y)\le \theta$3. If the primitive costs are $D(x,y)\le \theta$4 and $D(x,y)\le \theta$5, $D(x,y)\le \theta$6, the overall complexity is

$D(x,y)\le \theta$7

where $D(x,y)\le \theta$8 is the cost of creating singleton blocks (Stadtländer et al., 2021).

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

$D(x,y)\le \theta$9

In $I(x,y)\subseteq C$0, blocks are axis-aligned hyperrectangles encoded by minimum and maximum points, with total time

$I(x,y)\subseteq C$1

In $I(x,y)\subseteq C$2, each finite $I(x,y)\subseteq C$3-connected hull is a convex polygon with vertices from the input, yielding total time

$I(x,y)\subseteq C$4

The paper emphasizes that without the weak convexity constraint the corresponding problems are computationally intractable (Stadtländer et al., 2021).

An extensional variant is also given for finite graphs with geodesic distance. After precomputing all pair distances, it iteratively adds intervals $I(x,y)\subseteq C$5 whenever $I(x,y)\subseteq C$6. The total complexity is $I(x,y)\subseteq C$7 time and $I(x,y)\subseteq C$8 space, where $I(x,y)\subseteq C$9 is the preprocessing cost. Empirical tests on grid graphs, Delaunay triangulations, and random graphs are reported to achieve at least $\theta$0 accuracy in vertex classification tasks with $\theta$1–$\theta$2 labeled vertices (Stadtländer et al., 2021).

4. Weakly $\theta$3-convex sets in Euclidean space

In the Euclidean generalized-convexity literature, weakly $\theta$4-convex sets are defined by escaping affine planes rather than metric intervals. An open set $\theta$5 is weakly $\theta$6-convex if for every boundary point $\theta$7 there exists an $\theta$8-dimensional plane $\theta$9 with $m$00 and $m$01. A closed set $m$02 is weakly $m$03-convex if it is approximated from the outside by a nested family of open weakly $m$04-convex sets whose intersection is $m$05 (Osipchuk, 2021). A point $m$06 is an $m$07-nonconvexity point if every $m$08-plane through $m$09 intersects $m$10; the set of all such points is denoted $m$11 (Osipchuk, 2021).

This definition yields strong topological constraints. Osipchuk proves that any closed, weakly $m$12-convex set in $m$13 with non-empty set of $m$14-nonconvexity points consists of not less than three connected components (Osipchuk, 2021). In the planar case, if $m$15 is closed, weakly $m$16-convex, has finitely many components, and $m$17, then $m$18 is itself weakly $m$19-convex (Osipchuk, 2021). A later result states more generally that if a closed weakly $m$20-convex set in $m$21 has non-empty interior, then the interior is weakly $m$22-convex (Osipchuk, 2024).

The same literature establishes existence in the opposite direction: for each $m$23 and each $m$24, there exist open domains and closed connected sets in $m$25 that are weakly $m$26-convex but not $m$27-convex (Osipchuk, 2021). Thus the component lower bound for the extreme case $m$28 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 $m$29 that are weakly $m$30-convex but not $m$31-convex, with the inner triangle $m$32 serving as the set of $m$33-nonconvexity points. The closure $m$34 is then a closed, connected weakly $m$35-convex set in $m$36 that is not $m$37-convex (Osipchuk, 2021).

A further structural theorem for open weakly $m$38-convex sets states that if the set of $m$39-nonconvexity points is non-empty, then that set is itself open; the same statement holds for weakly $m$40-semiconvex sets and their nonsemiconvexity regions (Osipchuk, 2024). 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 $m$41 and $m$42, the modulus of nonconvexity is

$m$43

The set $m$44 is weakly convex with modulus $m$45 on $m$46 if for every $m$47 one has

$m$48

Equivalently, whenever $m$49, there exists $m$50 such that $m$51 (Balashov et al., 2010).

This notion generalizes Vial’s weak convexity in Hilbert spaces and is closely related to Clarke’s proximal smoothness. In a Hilbert space $m$52, if $m$53 with

$m$54

then the new definition is equivalent to Vial’s. In uniformly smooth Banach spaces, proximal smoothness at radius $m$55 implies $m$56 for all $m$57, and under the stronger inequality

$m$58

one recovers uniqueness of nearest-point projections and hence proximal smoothness (Balashov et al., 2010). Classical convexity is the degenerate case $m$59.

Several analytic consequences follow. If $m$60 is weakly convex with modulus $m$61 and

$m$62

then every point of the $m$63-neighborhood $m$64 has a unique nearest-point projection $m$65 (Balashov et al., 2010). Under an additional technical condition relating $m$66 and $m$67, the projection is uniformly continuous, with a Hölder-type modulus $m$68 as $m$69 (Balashov et al., 2010).

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 (Balashov et al., 2010). 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 $m$70 with geodesic distance $m$71, a subset $m$72 is weakly convex if for every $m$73 there exists a $m$74-$m$75 geodesic $m$76 such that all vertices of $m$77 lie in $m$78 (Anand et al., 26 Jan 2025). 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 $m$79, the minimum size of a dominating set $m$80 such that $m$81 is weakly convex (Anand et al., 26 Jan 2025).

The 2025 study of graph products derives bounds for $m$82 under the Cartesian, strong, and lexicographic products. For example, if $m$83 and $m$84, then

$m$85

and for the lexicographic product one has

$m$86

where $m$87 is defined from isolated vertices in a minimum outer $m$88-dominating set (Anand et al., 26 Jan 2025). These results show that weak convexity retains enough path structure to support product-graph domination theory.

The Euclidean weakly $m$89-convex literature also connects to “shadow” problems. Dakhil–Zelinskii–Klishchuk study the role of weakly $m$90-convexity in the geometry of shadows cast by balls and prove, among other results, a lower bound

$m$91

for the number of pairwise disjoint equal-radius balls needed to cast a tangent shadow at every point of the sphere (Dakhil et al., 2017). 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 $m$92-semiconvex sets replace escaping lines by escaping rays. Open weakly $m$93-semiconvex but non-$m$94-semiconvex sets with smooth boundary require at least four connected components, while closed weakly $m$95-semiconvex but non-$m$96-semiconvex sets require at least three connected components; these bounds are sharp in the cited constructions (Osipchuk, 2020, Osipchuk, 2017). This comparison is useful because weak $m$97-convexity and weak $m$98-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.