  In this article, we set up a variational problem to arrive at the equation of
the maximal hypersurface in the interior of a spherically symmetric evolving
trapped region. In the first part of the article, we present the Lagrangian and
the corresponding Euler-Lagrange equations that maximize the interior volume of
a trapped region that is formed dynamically due to infalling matter in
D-dimensions, with and without the cosmological constant. In the second part,
we explore the properties of special radii, which we call Reinhart radii, that
play a crucial role in approximating the maximal interior volume of a black
hole. We derive a formula to locate these Reinhart radii in terms of coordinate
invariants like area radius, principle values of the energy-momentum tensor,
Misner-Sharp mass, and cosmological constant. Based on this formula, we
estimate the location of Reinhart radii in various scenarios: (a) the case of
static BTZ black holes in (2 + 1)-dimensions and for the Schwarzschild,
Schwarzschild-de Sitter, and Schwarzschild-anti-de Sitter black holes in
D-dimensions. We plot the location of the Reinhart radii in relation to the
event horizon and cosmological horizon in a static D-dimensional scenario, (b)
cosmological case: we prove that these Reinhart radii do not exist for
homogeneous evolving dust for the zero and negative cosmological constant but
exist in the presence of positive cosmological constant when the scale factor
is greater than a critical value. We also show the relation between these
Reinhart radii and Kodama vectors.
