Bornivorous set

In functional analysis, a subset of a real or complex vector space formula_1 that has an associated vector bornology formula_2 is called bornivorous and a bornivore if it absorbs every element of formula_3 
If formula_1 is a topological vector space (TVS) then a subset formula_5 of formula_1 is bornivorous if it is bornivorous with respect to the von-Neumann bornology of formula_1. 
Bornivorous sets play an important role in the definitions of many classes of topological vector spaces (e.g. Bornological spaces). 
Definitions.
If formula_1 is a TVS then a subset formula_5 of formula_1 is called bornivorous and a bornivore if formula_5 absorbs every bounded subset of formula_12 
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 infrabounded if it maps Banach disks to bounded disks. 
A disk in formula_1 is called infrabornivorous 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 (i.e. is "compactivorous"). 
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 formula_14 is a vector subspace of finite codimension in a locally convex space formula_1 and formula_16 If formula_17 is a barrel (resp. bornivorous barrel, bornivorous disk) in formula_14 then there exists a barrel (resp. bornivorous barrel, bornivorous disk) formula_19 in formula_1 such that formula_21 
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 formula_1 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 formula_1 be formula_24 as a vector space over the reals. 
If formula_5 is the balanced hull of the closed line segment between formula_26 and formula_27 then formula_5 is not bornivorous but the convex hull of formula_5 is bornivorous. 
If formula_30 is the closed and "filled" triangle with vertices formula_31 and formula_27 then formula_30 is a convex set that is not bornivorous but its balanced hull is bornivorous.