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The Mathematical Properties of Varentropy Now we discuss some mathematical properties of varentropy at a discrete distribution with a fixed entropy. Let $X$ be a discrete random variable with support $\{x_1, x_2, \ldots, x_k\}$ and corresponding probabilities $P(X = x_i) = p_i$, and let $H(X) = H_0$ be the entropy of $X$. The varentropy $V(X)$ can be expressed as: $$ V(X) = \sum_{i=1}^{V} p_i \left( -\log(p_i) \right)^2 - H_0^2 $$Lagrange Multiplier Method To find the lower bound of varentropy for a fixed entropy $H_0$, we can use the method of Lagrange multipliers. We want to maximize $V(X)$ subject to the constraints that the probabilities sum to 1 and the entropy is fixed. We can consider it as the following optimization problem: ...