Operations with slicely countably determined sets

TL;DR

This paper investigates the stability of slicely countably determined (SCD) sets under various operations, providing counterexamples and clarifying the relationship between almost SCD and SCD sets in spaces with the Daugavet property.

Contribution

It shows that SCD property is not preserved under union, intersection, or Minkowski sum, and that almost SCD sets need not be SCD, answering open questions in the field.

Findings

SCD sets are not preserved under union, intersection, or Minkowski sum.

Examples of SCD sets exist in every space with the Daugavet property.

Almost SCD sets can fail to be SCD.

Abstract

The notion of slicely countably determined (SCD) sets was introduced in 2010 by A.~Avil\'{e}s, V.~Kadets, M.~Mart\'{i}n, J.~Mer\'{i} and V.~Shepelska. We solve in the negative some natural questions about preserving being SCD by the operations of union, intersection and Minkowski sum. Moreover, we demonstrate that corresponding examples exist in every space with the Daugavet property and can be selected to be unit balls of some equivalent norms. We also demonstrate that almost SCD sets need not be SCD, thus answering a question posed by A. Avil\'{e}s et al.