What do we call these corrugated board sleeves put upon cardboard boxes to provide additional strength?

Question

This kind of a sleeve from cardboard put upon a cardboard box is called обечайка (obechaika) in Russian. What do we call them in English? I was told on Proz.com that we call them "cardboard shells", is that so?

I was translating a text in which medical products were packaged into cardboard boxes with such additional shells to provide additional protection.

Such shells provide additional strength but sometimes such shells are made from decorative paper with fancy design and are put around, for instance, a box of chocolates to make it look more refined. These are also obechaikas in Russian.

Here, a cardoard box is slid inside an obechaika:

Here's another use of the term obechaika, a decoratice paper shell around a food product in a plastic container:

In this usage obechaika is more frequently called shuber, for some reason. The word is yet absent from Russian dictionaries, might be a new coinage.

Answer 1

I would use “cardboard sleeve”, and it would apply to both your examples.