We investigate a novel setting for polytope rigidity, where a flex must preserve edge lengths and the planarity of faces, but is allowed to change the shapes of faces. For instance, the regular cube is flexible in this notion. We present techniques for constructing flexible polytopes and find that flexibility seems to be an exceptional property. Based on this observation, we introduce a notion of generic realizations for polytopes and conjecture that convex polytopes are generically rigid in dimension $d\geq 3$. We prove this conjecture in dimension $d=3$. Motivated by our findings we also pose several questions that are intended to inspire future research into this notion of polytope rigidity.

翻译：我们研究了一种新颖的多面体刚性设定，其中形变必须保持边长和面的共面性，但允许改变面的形状。例如，在这种定义下，规则立方体是可形变的。我们提出了构造可形变多面体的技术，并发现可形变性似乎是一种例外性质。基于这一观察，我们引入了多面体的泛型实现概念，并猜想在维度 $d\geq 3$ 中凸多面体是泛型刚性的。我们在维度 $d=3$ 时证明了这一猜想。受研究结果启发，我们还提出了若干问题，旨在推动未来对这一多面体刚性概念的进一步研究。