Bornivorous set

In functional analysis, a subset of a real or complex vector space $$X$$ that has an associated vector bornology $$\mathcal{B}$$ is called bornivorous and a bornivore if it absorbs every element of $$\mathcal{B}.$$ If $$X$$ is a topological vector space (TVS) then a subset $$S$$ of $$X$$ is bornivorous if it is bornivorous with respect to the von-Neumann bornology of $X$.

Bornivorous sets play an important role in the definitions of many classes of topological vector spaces, particularly bornological spaces.

Definitions
If $$X$$ is a TVS then a subset $$S$$ of $$X$$ is called ' and a ' if $$S$$ absorbs every bounded subset of $$X.$$

An absorbing disk in a locally convex space is bornivorous if and only if its Minkowski functional is locally bounded (i.e. maps bounded sets to bounded sets).

Infrabornivorous sets and infrabounded maps
A linear map between two TVSs is called  if it maps Banach disks to bounded disks.

A disk in $$X$$ is called  if it absorbs every Banach disk.

An absorbing disk in a locally convex space is infrabornivorous if and only if its Minkowski functional is infrabounded. A disk in a Hausdorff locally convex space is infrabornivorous if and only if it absorbs all compact disks (that is, if it is "").

Properties
Every bornivorous and infrabornivorous subset of a TVS is absorbing. In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.

Two TVS topologies on the same vector space have that same bounded subsets if and only if they have the same bornivores.

Suppose $$M$$ is a vector subspace of finite codimension in a locally convex space $$X$$ and $$B \subseteq M.$$ If $$B$$ is a barrel (resp. bornivorous barrel, bornivorous disk) in $$M$$ then there exists a barrel (resp. bornivorous barrel, bornivorous disk) $$C$$ in $$X$$ such that $$B = C \cap M.$$

Examples and sufficient conditions
Every neighborhood of the origin in a TVS is bornivorous. The convex hull, closed convex hull, and balanced hull of a bornivorous set is again bornivorous. The preimage of a bornivore under a bounded linear map is a bornivore.

If $$X$$ is a TVS in which every bounded subset is contained in a finite dimensional vector subspace, then every absorbing set is a bornivore.

Counter-examples
Let $$X$$ be $$\mathbb{R}^2$$ as a vector space over the reals. If $$S$$ is the balanced hull of the closed line segment between $$(-1, 1)$$ and $$(1, 1)$$ then $$S$$ is not bornivorous but the convex hull of $$S$$ is bornivorous. If $$T$$ is the closed and "filled" triangle with vertices $$(-1, -1), (-1, 1),$$ and $$(1, 1)$$ then $$T$$ is a convex set that is not bornivorous but its balanced hull is bornivorous.