  In this paper we continue the study of symplectically self-polar convex
bodies started in arXiv:2211.14630. We construct symplectically self-polar
convex bodies of the minimal Ekeland-Hofer-Zehnder capacity. This in turn
proves that the lower bound for the Ekeland-Hofer-Zehnder capacity for
centrally symmetric convex bodies obtained in arXiv:1801.00242 cannot be
improved. We also make some numerical experiments and speculations regarding
the minimal volume of symplectically self-polar convex bodies.
