We show that for any compact convex set $K$ in $mathbb{R}^d$ and any finite family $mathcal{F}$ of convex sets in $mathbb{R}^d$, if the intersection of every sufficiently small subfamily of $mathcal{F}$ contains an isometric copy of $K$ of volume $1$
, then the intersection of the whole family contains an isometric copy of $K$ scaled by a factor of $(1-varepsilon)$, where $varepsilon$ is positive and fixed in advance. Unless $K$ is very similar to a disk, the shrinking factor is unavoidable. We prove similar results for affine copies of $K$. We show how our results imply the existence of randomized algorithms that approximate the largest copy of $K$ that fits inside a given polytope $P$ whose expected runtime is linear on the number of facets of $P$.
Tverberg-type theory aims to establish sufficient conditions for a simplicial complex $Sigma$ such that every continuous map $fcolon Sigma to mathbb{R}^d$ maps $q$ points from pairwise disjoint faces to the same point in $mathbb{R}^d$. Such results a
re plentiful for $q$ a power of a prime. However, for $q$ with at least two distinct prime divisors, results that guarantee the existence of $q$-fold points of coincidence are non-existent---aside from immediate corollaries of the prime power case. Here we present a general method that yields such results beyond the case of prime powers. In particular, we prove previously conjectured upper bounds for the topological Tverberg problem for all $q$.
