It is known from Blaschke and later Sas that any convex region $P$ with area $1$ admits an inscribed triangle with area at least $\frac{3\sqrt{3}}{4\pi}$. What if we require the triangle to have a given extreme point of P as vertex? The optimal constant should be between $\frac{1}{4}$ and $\frac{1}{3}$, but I can't tighten these bounds by too much.

It is known from Blaschke and later Sas that any convex region $P$ with area $1$ admits an inscribed triangle with area at least $\frac{3\sqrt{3}}{4\pi}$. What if we require the triangle to have a given boundary point of P as vertex? The optimal constant should be between $\frac{1}{4}$ and $\frac{1}{3}$, but I can't tighten these bounds by too much.

It is known from Blaschke and later Sas that any convex region $P$ with area $1$ admits an inscribed triangle with area at least $\frac{3\sqrt{3}}{4\pi}$. What if we require the triangle to have a given point as vertex? The optimal constant should be between $\frac{1}{4}$ and $\frac{1}{3}$, but I can't tighten these bounds by too much.

It is known from Blaschke and later Sas that any convex region $P$ with area $1$ admits an inscribed triangle with area at least $\frac{3\sqrt{3}}{4\pi}$. What if we require the triangle to have a given extreme point of P as vertex? The optimal constant should be between $\frac{1}{4}$ and $\frac{1}{3}$, but I can't tighten these bounds by too much.