Absolutely convex set

In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk.

Version

v1 · 2026-09-28 · History

Domain-specific #

7827

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Functional Analysis, Topological Vector Spaces → Mathematics

Core Idea

Absolutely convex set is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead.

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Centered No-Dent Shapes

Picture a shape drawn around a center dot, with no dents or holes. Take any point inside and slide it toward the center dot, or flip it to the exact opposite side of the center: it's still inside. A shape like that, such as a circle centered on the dot, is called absolutely convex.

Balanced No-Dent Shapes

In math, a shape is 'convex' if a straight line between any two of its points stays completely inside — no dents or holes. A shape is 'balanced' if, whenever a point is in it, shrinking that point toward the center, or flipping it to the opposite side of the center, keeps it inside. A set that is both convex and balanced is called absolutely convex, or a 'disk'. A filled circle or square centered on the origin is an example; a triangle with a corner at the center isn't, because flipping it through the center takes points outside.

Convex and Balanced Sets

In a real or complex vector space, a subset C is convex if it contains the line segment between any two of its points, and balanced (or 'circled') if sC ⊆ C for every scalar s with |s| ≤ 1, meaning scaling toward the origin, and for real spaces flipping through it, keeps you inside. A set that is both convex and balanced is called absolutely convex, disked, or simply a disk. Equivalently, rc + sd is in C whenever c, d are in C and |r| + |s| ≤ 1. The disked hull (absolute convex hull) of a set is the intersection of all disks containing it. A useful consequence: for a disk D and scalars r, s, sD = |s|D and (rD) ∩ (sD) = min(|r|, |s|)·D.

 

A subset C of a real or complex vector space is absolutely convex (disked) if it is convex and balanced, i.e., sC ⊆ C whenever |s| ≤ 1; such a set is called a disk. Equivalently, rc + sd ∈ C whenever c, d ∈ C and |r| + |s| ≤ 1. The absolutely convex (disked) hull of a set is the intersection of all disks containing it. For a disk D and scalars r, s: sD = |s|D and (rD) ∩ (sD) = min{|r|, |s|}D. Replacing the exponent condition by |r|^p + |s|^p ≤ 1 yields the related p-convex notions. Order matters when building hulls: the balanced hull of a convex set need not be convex — the balanced hull of a horizontal segment is an 'hourglass' of two triangles meeting at the origin. In a topological vector space, bounded disks have useful properties: for a sequence xᵢ in a bounded disk, the partial sums of Σ 2^(−i) xᵢ are Cauchy.

Scope of Application

• Generalizations. A is any non-negative function q : X \to \R that satisfies the following conditions.

• Generalizations. For example, whenever 0 then the map q(f) = \int{\R} |f(t)|^p d t used to define the Lp space Lp(\R) is a p -seminorm but not a.

• Definition. A subset S of a real or complex vector space X is called a ' and is said to be ', ', and ' if any of the following equivalent conditions is satisfied.

• Definition. for any scalars a and b, if |a| + |b| \leq 1 then a S + b S \subseteq S.

• Definition. for all scalars a, b, and c, if |a| + |b| \leq |c|, then a S + b S \subseteq c S.

Clarity

A clear use of Absolutely convex set names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called.

Manages Complexity

Absolutely convex set compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—the disked hull of S will be denoted by \operatorname{disk} S or \operatorname{cobal} S and it is equal to each of the following sets.—and the practical consequence—for all scalars a, b, and c, if |a| + |b| \leq |c|, then a S + b S \subseteq.

Abstract Reasoning

1. Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
2. State the relation. Use the source-grounded identity: In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk.
3. Check operation and conditions.

Knowledge Transfer

Within the home domain. Knowledge about Absolutely convex set transfers literally when a new case preserves the same carrier type, relation, and recognition test. A is any non-negative function q : X \to \R that satisfies the following conditions. For example, whenever 0 then the map q(f) = \int{\R} |f(t)|^p d t used to define the Lp space Lp(\R) is a p -seminorm but not a seminorm. Beyond the home domain. No canonical parent is asserted for Absolutely convex set.

Neighborhood in Abstraction Space

Absolutely convex set sits in a moderately populated region (47^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Topological & Functional-Analytic Spaces (13 abstractions)

Nearest neighbors

• Balanced set — 0.88
• Affine hull — 0.88
• Infrabarrelled space — 0.86
• DF-space — 0.86
• Convex hull — 0.86

Computed from structural-signature embeddings · 2026-10-08