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Let $P$ be a convex polytope in Euclidean space $\mathbf{R}^3$ and $\Gamma$ be a closed curve which passes through all vertices of $P$. How small can the length $L$ of $\Gamma$ be? More specifically, …

Mark Levi's book The Mathematical Mechanic: Using Physical Reasoning to Solve Problems is full of concrete examples of applying physical intuition in geometry, including even a proof of Gauss Bonnet …

In another paper with James Wenk, we have shown that the condition in Zalgaller's conjecture that the curve lie outside the sphere is not necessary, that is, the inequality $L\geq 4\pi$ holds for all …

In a recent paper, Centers of disks in Riemannian manifolds, Igor Belegradek and I study whether it is possible to extend to nonconvex objects the notion of center of mass or other classical centers a …

Anton's construction depends continuously on the curve but does not seem quite canonical, in the sense that smoothings of two congruent curves may not be congruent. We can ensure that this will be t …

Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$. If the curves are sm …

There is a natural length-preserving operation which transforms any rectifiable space curve $\gamma\colon [a,b]\to R^n$ into a planar curve $\tilde\gamma \colon [a,b]\to R^2$. This operation, which ha …

For $n=3$ the answer is yes, as was shown by Fejes Tóth in 1943; see the Theorem on p.34 of his book Regular Figures. For $n=4$ the answer is also positive as shown in the 2000 paper, The blocking num …

Let $v_i$ be $2n$ points in $\mathbf{R}^n$, with equal distance $|v_i|$ from the origin. Suppose that the convex hull of these points contains the unit ball. Is it known that $|v_i|\geq\sqrt{n}$? Pre …

In 1926 Kneser showed that homeomorphisms of $\mathbf{S}^2$ admit a retraction into the orthogonal group $O(3)$. Smale extended this result to Diffeomorphisms of $\mathbf{S}^2$ in 1958; however, in th …

James Wenk and I just finished a paper proving Zalgaller's sphere inspection conjecture for closed curves: Shortest closed curve to inspect a sphere. We show that in $R^3$ any closed curve $\gamma$ w …

In any Hadamard space, projection into convex sets is non-expansive; see Proposition 2.4(4) in Metric spaces of non-positive curvature by Bridson and Haefliger.

I suppose that we have a fixed contour $\Gamma$ which is a graph over $\partial D$, and you want to show that the minimal graph spanned by $\Gamma$ is area minimizing. First, by the maximum principle …

Suppose that we have a convex cap, i.e., a convex surface in $R^3$ homeomorphic to a disk whose boundary lies in a plane. Reflect the cap through the plane of its boundary and glue it back to the orig …

One of the fundamental facts in fiber bundle theory is the following result for existence and extension of sections (see Thm. 9 in this paper of Palais, and compare with Thm. 12.2 in Steenrod's book) …