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Take a path that joins the antipodes and concatenate it with its symmetric image. Get a centrally symmetric closed path on the boundary of the cube. If this path avoids one of the facets of the cube ( …

The Dehn functional of a polyhedron $P$ with edge lengths $\ell_i$ and exterior dihedral angles $\theta_i$ is $D(P) = \sum_i \ell_i \otimes_{\mathbb Q} (\theta_i\, \mathrm{mod}\, \mathbb{Q}\pi)$. It l …

Answer to question 1: proper compact subsets of convex surfaces are infinitesimally flexible. In his book Pogorelov proves infinitesimal flexibility of the graph of a convex function over a strictly …

Yes, this is true, and you are right, this follows from a generalization of the Brunn-Minkowski inequality. Let $K_s = \{x \mid f(x) \le s\}$, so that $M_s = \partial K_s$. We have $K_s \supseteq (1- …

There is definitely no counterexample for convex polyhedra and no counterexample for bodies with smooth boundary with non-degenerate second fundamental form. (In the latter case even small open subset …

In the first note to section 4.2 of Schneider, Rolf, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and Its Applications. 44. Cambridge: Cambridge University Press. xiii, 490 …

Yes, this is called the principal kinematic formula: $$\int \chi(K \cap gL)\, dg = \sum_{k=0}^n c_{nk} V_k(K) V_{n-k}(L),$$ where $V_i$ are the intrinsic volumes, and $c_{nk}$ certain constants. See e …

There is no simple description. A face of the Minkowski sum is the Minkowski sum of faces of the summands. More exactly, if $F_u(P)$ denotes the face of $P$ with outer normal $u$, then $$ F_u(P+Q) = …

A convenient way to think about it is to represent a convex body in terms of its support function (restricted to the unit sphere). Minkowski addition corresponds to the addition of support functions. …

No. A subdivision that can be lifted to a convex cap is called regular (or coherent, or weighted Delaunay). Here is an example of a non-regular subdivision: For more on this, I recommend the book "T …

The following answers questions 1 to 4. The misfit and the oriented misfit can be arbitrarily large. Let $A$ be a ball, and let $B$ be a very long (and thin so that it has volume $1$) cylinder. We ha …

The existence of a symmetric embedding can be proved as follows. Approximate the metric by a sequence of symmetric hyperbolic cone-metrics (i. e. locally hyperbolic with cone points of angles $< 2\pi …

First, a linear relation can involve only volumes of faces of equal dimensions because scaling a polytope by $\lambda$ multiplies the volume of a $k$-face by $\lambda^k$. For a polytope in $\mathbb{R …

I doubt that much can be said about the shape of a convex body given its intrinsic volumes (this is a more common name for the basic valuations). Intuitively, this is because the space of convex bodie …

Schoenhardt polyhedron (wikipedia) is a star-shaped polyhedron in $\mathbb{R}^3$ with triangular faces that cannot be triangulated without subdividing its faces. So the answer is no even without requi …