57 research outputs found
On diversities and finite dimensional Banach spaces
A diversity in is a function defined over every finite set of
points of mapped onto , with the properties that
if and only if and , for every finite sets with . Its importance
relies in the fact that, amongst others, they generalize the notion of metric
distance.
We characterize when a diversity defined over , , is
Banach-embeddable, i.e. when there exist points , , and a
symmetric, convex, and compact set such that
, where
denotes the circumradius of with respect to . Moreover, we also
characterize when a diversity is a Banach diversity, i.e. when
, for every finite set , where is an
-dimensional, symmetric, convex, and compact set
Narrowing the gaps of the missing Blaschke-Santal\'o diagrams
We solve several new sharp inequalities relating three quantities amongst the
area, perimeter, inradius, circumradius, diameter, and minimal width of planar
convex bodies. As a consequence, we narrow the missing gaps in each of the
missing planar Blaschke-Santal\'o diagrams. Furthermore, we extend some of
those sharp inequalities into higher dimensions, by replacing either the
perimeter by the mean width or the area by the volume
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