Suppose that the convex hull of the Minkowski sum of several compact connected sets in $\mathbb R^d$ contains the unit ball centered at the origin and the diameter of each set is less than $\delta$. If $\delta$ is very small (this smallness may depend on $d$ but on nothing else), does it follow that the sum itself contains the origin?
[Math] Minkowski sum of small connected sets
convexitymg.metric-geometry
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Possibly I misunderstood your question, but I do not see a hope for a suitable characterization because of the following example (inspired by the comment of Bill Johnson): Consider a set $A$ in $\ell_2^2$ ($2$-dimensional Euclidean space) consisting of the union of the open unit ball and some subset of the unit sphere. The set $A$ is always $\sigma$-convex. On the other hand $A$ is (this can be shown directly) homeomorphic to a subset $O$ consisting of the union of the open unit ball of $\ell_1^2$ and a suitable subset of the sphere. The set $O$ is $\sigma$-convex (and even convex) in some exceptional cases only. Unless the set $A$ contains at most one point on the boundary, the homeomorphism can be modified in such a way that the (modified) $O$ is not $\sigma$-convex.
I misread the problem as that you just wanted the origin in the convex hull. This is a nice problem whose solution deserves to be better known, and this provides an upper bound for the probability that the convex hull contains the unit ball, and the limiting behavior as $r \to \infty$.
J. G. Wendel; "A Problem in Geometric Probability," Mathematica Scandinavica 11 (1962) 109-111. Zbl 108.31603
A special case ($4$ points on the $2$-sphere) was A6 on the 1992 Putnam exam.
You say both $d$-dimensional sphere and $d$-dimensional ball, though I would say the unit $3$-dimensional ball is bounded by a $2$-sphere. I'll assume you mean the ball in $\mathbb{R}^d$.
Consider the kernel of the map $\mathbb{R}^n \to \mathbb{R}^{d}$ of linear combinations of the points. Generically, this kernel has dimension $n-d$. The origin is contained in the convex hull precisely when this kernel intersects the positive orthant, since then some linear combination can be rescaled to have total weight $1$.
By symmetry, all orthants are hit equally often. (Negating a point reflects the kernel, and the action this generates is transitive on the orthants.) A generic $a$-dimensional subspace of $\mathbb{R}^{n}$ hits $2\sum_{i=0}^{a-1} {n-1 \choose i}$ out of $2^{n}$ orthants, so the probability that the convex hull of $n$ random points contains the origin is
$$2^{-n+1}\sum_{i=0}^{n-d-1} {n-1\choose i}. $$
This is an upper bound for the probability that the unit ball is contained in the convex hull.
The origin is contained in the convex hull with probability $1/2$ when the kernel and its orthogonal complement have the same dimension, when $n=2d$.
Best Answer
I finally figured it out. My solution is here. I would repost it on mathoverflow but until LaTeX is enabled, it is quite hard for me to communicate such things here...