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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 …

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 …

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 …

I have recently finished a paper called The length, width, and inradius of space curves where it is shown that the length $L$ of any closed curve $\gamma\colon[a,b]\to \mathbf{R}^3$ inspecting the u …

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 …

The baseball stitches curve suggested by Gjergji Zaimi appears in another paper of Zalgaller: V. A. Zalgaller. Extremal problems on the convex hull of a space curve. Algebra i Analiz, 8(3):1–13, 1996 …

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 …

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) …

The conjecture attributed to Hadamard, if one regards that as being concerned with the existence of a complete embedded negatively curved surface in a ball, is still open. I have read the correspondin …

The answer to Question 1 is yes, which is precisely Lemma 3.6 in the paper: Boundary torsion and convex caps of locally convex surfaces, J. Differential Geom., 105 (2017), 427-486. Although the lem …

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.

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 …

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 …

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 …