The simplex was conjectured to be the extremal convex body for the two following problems of asymmetry: (P1) What is the minimal possible value of the quantity $$\max _{K'} |K'|/|K|$$maxK?|K?|/|K|? Here, $$K'$$K? ranges over all symmetric convex bodies contained in K. (P2) What is the maximal possible volume of the Blaschke body of a convex body of volume 1? Our main result states that (P1) and (P2) admit precisely the same solutions. This complements a result from Boroczky et al. (Discrete Math 69:101---120, 1986), stating that if the simplex solves (P1), then the simplex solves (P2) as well.