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This isodiametric hull is not well defined. The first observation is that any curve of constant width is its own isodiametric hull. Now let's take an isosceles triangle ABC with $\angle A=20^{\circ}$, …

Given $e$ so that it satisfies your condition of distinct triangles intersecting nontrivially we will prove that it comes from an associated polygon by induction on the number of vertices of the polyg …

This is still in terms of trigonometry, but it's pretty quick. Let $\vec{u}$ and $\vec{v}$ be the unit vectors in the direction of $\vec{bc}$ and $\vec{ba}$, respectively. We have $$\angle dba + \angl …

I believe this problem has been mentioned a few times in the literature, and has been solved for certain restrictions on the curve. For example if the curve has no four coplanar points then the maxima …

Convexity is equivalent to the function $r(\theta):[0,2\pi)\to\mathbb{R}^2$ being well-defined and satisfying the condition $$| r(\lambda \theta _1+(1-\lambda)\theta _2)| \geq \left| \lambda r(\theta …

Consider the $k-1$ dimensional simplex given by $\alpha_1+\alpha_2+\cdots \alpha_k=1, \alpha_i\geq 0$. The equations $e_i\cdot (\sum_{j=1}^k \alpha_j x_j)=0$ for $1\le i\le n$ describe $n$ hyperplanes …

Let $M$ be the circulant matrix whose rows are given by cyclic shifts of $(v_1,\dots v_d)$ and let $P(x)=v_1+v_2x+\cdots+v_dx^{d-1}$ be the associated polynomial. Moreover, let $s$ be the degree of $\ …