I solved H using an $$$O(n)$$$ solution. Simply said, the task can be modeled as an LP task with $$$4$$$ variables and $$$2n$$$ constraints. Thus, we can use methods to solve low-dimensional LP in linear time. Combining Seidel's method and Clarkson's method, this can be done in $$$O(d^2 n + \text{the rest is not really important whatsoever in low dimensions})$$$. When some paper says that low-dimensional LP can be done in $$$O(n)$$$, they're usually pointing towards this. But yet, builtin floating point types are not the best for numerical stability, so my solution's implementation (shamelessly stolen from dacin21's codebook) uses bigints and rational arithmetic. And then, this turns out to be slower than the ternary-search solution (but just enough to fit the TL).

I have not tested whether this works well on higher dimensions, but I believe it should be fine for at most $$$5$$$ variables in the LP formulation. If we have more than $$$5$$$ variables, then I would rather pray for the Simplex algorithm to work very well.