Description
The concept of reducedness was defined by Heil in 1978 in hope of tackling volume minimizing problems concerning the minimal width of the body.
A convex body is called reduced, if for any different convex body that is contained within, the minimal width is strictly smaller.
At first, the definition suggests that reduced bodies are some kind of dual of bodies of constant width, but in fact it is a broader family.
A classical result of Pál from 1921 states that the regular triangle minimizes the area if the minimal width is fixed. Lassak conjectured that the area among reduced bodies of a fixed minimal width is maximized by the circle and the quarter of a disk, and he proved that all reduced polygons have a smaller area.
Similar investigations have been made on the sphere, but the concept of reducedness in the hyperbolic space is very recent as the formerly known hyperbolic width functions did not make a good fit for this purpose.
After introducing a new concept of hyperbolic width by Lassak, we will discuss some unexpected facts on hyperbolic reduced bodies including Pál's problem, we will see how hyperbolic reduced polytopes can differ from Euclidean or spherical ones, and we will answer four of Lassak's recent questions.
This talk is partially based on joint works with Károly Jr. Böröczky, András Csépai and Ansgar Freyer.