This is more of a followup to Noam's answer. Vilcu had continued to work on the subject, see this paper by him, Rouel and Itoh: https://www.evernote.com/shard/s24/sh/a5f3c60d-7c69-4a02-81db-e5721253977d/a51556c9763fe199c9981c13d950b97c

However, the most relevant papers seem to be those by Joel Rouel (unfortunately my university does not subscribe to "Advances in Geometry", so I can't tell with certainty what he does...):

MR2660417 (2011g:52021) Reviewed Rouyer, Joël On antipodes on a convex polyhedron II. (English summary) Adv. Geom. 10 (2010), no. 3, 403–417. 52B10 (51M04 52A15) More links PDF Clipboard Journal Article Make Link

This paper is a sequel to the author's paper [Part I, Adv. Geom. 5 (2005), no. 4, 497–507; MR2174479 (2006h:52005)]. Steinhaus asked about convex surfaces for which every point has precisely one antipode and the antipodal map is an involution. It is open whether the boundary of a convex polyhedron in R3 can satisfy that condition. This paper gives several relevant results about antipodes on convex polyhedra. For example: If a convex polyhedron has a small enough angle at some vertex, then some point has at least two antipodes. For every convex polyhedron, the antipodal map is not a local isometry. If P is a centrally symmetric convex polyhedron, then there is a finite union G⊂P of algebraic arcs such that each point p∈P∖G has a single antipode, which is not −p; hence no centrally symmetric convex polyhedron satisfies Steinhaus's conditions. Reviewed by Margaret M. Bayer

This is more of a followup to Noam's answer. Vilcu had continued to work on the subject, see this paper by him, Rouyer and Itoh: https://www.evernote.com/shard/s24/sh/a5f3c60d-7c69-4a02-81db-e5721253977d/a51556c9763fe199c9981c13d950b97c

However, the most relevant papers seem to be those by Joel Rouyer (unfortunately my university does not subscribe to "Advances in Geometry", so I can't tell with certainty what he does...):

MR2660417 (2011g:52021) Reviewed Rouyer, Joël On antipodes on a convex polyhedron II. (English summary) Adv. Geom. 10 (2010), no. 3, 403–417. 52B10 (51M04 52A15) More links PDF Clipboard Journal Article Make Link

This paper is a sequel to the author's paper [Part I, Adv. Geom. 5 (2005), no. 4, 497–507; MR2174479 (2006h:52005)]. Steinhaus asked about convex surfaces for which every point has precisely one antipode and the antipodal map is an involution. It is open whether the boundary of a convex polyhedron in R3 can satisfy that condition. This paper gives several relevant results about antipodes on convex polyhedra. For example: If a convex polyhedron has a small enough angle at some vertex, then some point has at least two antipodes. For every convex polyhedron, the antipodal map is not a local isometry. If P is a centrally symmetric convex polyhedron, then there is a finite union G⊂P of algebraic arcs such that each point p∈P∖G has a single antipode, which is not −p; hence no centrally symmetric convex polyhedron satisfies Steinhaus's conditions. Reviewed by Margaret M. Bayer