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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 …

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 …

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 …

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 ( …

No, this equation is false for non-convex polyhedra. Take a cube and remove from inside of it a smaller cube. The resulting body has the same mean width (caliper diameter), but the sum of angles times …

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 …

As j.c. mentioned in his answer, the average distance between parallel supporting planes is better known as the mean width. More generally, one can take a convex body in $\mathbb{R}^n$ and consider th …

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 …

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 …

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 …

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. …

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 …

Yes, the mean width is monotone with respect to the length of any edge of a tetrahedron. This follows from these two facts: The mean width of a 3-dimensional polyhedron is proportional to $\sum_i \el …

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) = …