# Questions tagged [convex-geometry]

A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ...

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Let $\Omega$ be a complete metrizable space $\mathscr A$ its Borel $\sigma$-algebra and $\mathscr M$ the set of all probability measures on $\Omega$. Every non-empty subset $\mathscr P \subset \... 1answer 26 views ### Naming convention: looking for better terminology for “centrally symmetric smooth strictly convex bodies” I have recently found myself researching a certain type of convex body in$\mathbb{R}^2$, namely centrally symmetric smooth strictly convex bodies. Instead of repeating such a sentence repetitively I ... 0answers 76 views ### Upper bound$\tau_C := \int_{\|x\| \le 1}(vol(C \cap (x + C))/vol(C))dx$for a convex body$C \subseteq \mathbb R^n$, by reducing to a ball Let$C$be a convex body in$\mathbb R^n$, i.e a bounded convex subset of$\mathbb R^n$which has nonempty interior, and which is (A) open, or (B) closed (I'm not sure one makes more sense; choose the ... 0answers 40 views ### What do you call$\operatorname{diam} (A)^d/\mathcal{L}^d (A)$for$A \subseteq \mathbb{R}^d$convex? If$A \subseteq \mathbb{R}^d$is convex, is there a more or less established name for the quantity $$\operatorname{diam} (A)^d/\mathcal{L}^d (A),$$ where$\mathcal{L}^d (A)$is the Lebesgue measure of ... 0answers 41 views ### Separability of Minkowski Sum of well-behaved sets Let$A$and$C$be non-empty simply connected and connected subsets of$\mathbb{R}^k$and suppose that$C$is convex. Then is the Minkowski sum$A+C$separable? 0answers 53 views ### Polytopes with large dihedral angles The regular$d$-simplex has dihedral angle$\arccos(1/d)<90^\circ$, and the$d$-cube has dihedral angle exactly$90^\circ$. The maximal dihedral angle of a prism over a$(d-1)$-simplex is also$90^\...
Consider a closed, bounded and convex set $C \subset \mathbb{R}^{2}$ and denote its boundary with $\partial C$. It is very well-known that the Minkowski sum of two convex sets is convex again. What ...