A seminal theorem of Tverberg states that any set of \(T(r,d)=(r-1)(d+1)+1\) points in \({\mathbb {R}}^{d}\) can be partitioned into r subsets whose convex hulls have non-empty r-fold intersection. Almost any collection of fewer points in \({\mathbb {R}}^{d}\) cannot be so divided, and in these cases we ask if the set can nonetheless be P(r, d)-partitioned, i.e., split into r subsets so that there exist r points, one from each resulting convex hull, which form the vertex set of a prescribed convex d-polytope P(r, d). Our main theorem shows that this is the case for any generic \(T(r,2)-2\) points in the plane and any \(r\ge 3\) when \(P(r,2)=P_{r}\) is a regular r-gon, and moreover that \(T(r,2)-2\) is tight. For higher dimensional polytopes and \(r=r_{1}\cdots r_{k}\), \(r_{i} \ge 3\), this generalizes to \(T(r,2k)-2k\) generic points in \({\mathbb {R}}^{2k}\) and orthogonal products \(P(r,2k)=P_{r_{1}}\times \cdots \times P_{r_{k}}\) of regular polygons, and likewise to \(T(2r,2k+1)-(2k+1)\) points in \({\mathbb {R}}^{2k+1}\) and the product polytopes \(P(2r,2k+1)=P_{r_{1}}\times \cdots \times P_{r_{k}}\times P_{2}\). As with Tverberg’s original theorem, our results admit topological generalizations when r is a prime power, and, using the “constraint method” of Blagojević, Frick, and Ziegler, allow for dimensionally restricted versions of a van Kampen–Flores type and colored analogues in the fashion of Soberón.

特维尔贝格（Tverberg）的一个开明定理指出，\（T（r，d）=（r-1）（d + 1）+1 \）的任何集合都指向\（{\ mathbb {R}} ^ {d} \）可以划分为r个子集，这些子集的凸包具有非空的r折交集。\（{\ mathbb {R}} ^ {d} \）中几乎所有较少点的集合都不能如此划分，在这种情况下，我们询问集合是否仍可以进行P（r，  d）划分，即分成r个子集，以便存在r个点（每个所得凸包中的一个），这些点形成指定的凸d-多面体P（r，  d）。我们的主要定理表明这是任何通用的情况下\（T（R，2）-2 \）在平面和任何点\（R \ GE 3 \）时\（P（R，2）= P_ { r} \）是一个正则r -gon，而且\（T（r，2）-2 \）是紧的。对于高维多曲面和\（r = r_ {1} \ cdots r_ {k} \），\（r_ {i} \ ge 3 \），这一般化为\（T（r，2k）-2k \）泛型在点\（{\ mathbb {R}} ^ {2K} \）和正交产品P_ {R_ {1}} \倍\ cdots \倍\（P（R，2K）= P_ {R_ {K}} \ ）的规则多边形，同样以\（T（2r，2k + 1）-（2k + 1）\）点\（{{mathbb {R}} ^ {2k + 1} \）和乘积多峰\（P（2r，2k + 1）= P_ {r_ {1}} \ times \ cdots \ times P_ {r_ {k }} \ P_ {2} \）。与特维尔伯格的原始定理一样，当r为素数幂时，我们的结果也可以接受拓扑概括，并且使用Blagojević，Frick和Ziegler的“约束方法”，可以对van Kampen–Flores类型和彩色类似物进行尺寸限制Soberón的时尚。