Leave $ K $ be a politician in $ mathbb R ^ d $, exploit it by a factor $ lambda> 0 $. For a unitary vector. $ u in mathbb S ^ {d-1} $, $ lambda K $ has 2 support hyperplanes $ H_1 $ Y $ H_2 $ with its corresponding normal vectors towards the outside. $ u $ Y $ -u $. I only take into account $ u $It is such that $ H_1 $ Y $ H_2 $ they have a "good" position, which says they are not parallel to any face of $ K $, or equivalently, contain a single vertex of $ lambda K $.

Consider another parallel hyperplane. $ H_t $ such that the distance between him and $ H_1 $ is $ t $ (Of course $ t $ is smaller than the width of $ lambda K $ in the direction $ u $). So $ H_1 $ Y $ H_t $ define a limit of $ lambda K $.

My question is:

Is there a possibility that we can estimate the volume of that limit, in terms of $ u, lambda $ Y $ t $?

The easiest case is when $ t $ is smaller than the distance from the nearest vertex $ lambda K $ to $ H_1 $, then the volume is $ frac {1} {d} in ^ d $, where $ a $ is a constant depends on $ u $.

I mean, in general, it is very difficult to calculate the volume of capitals defined in a similar way to $ K $. But if we explode $ K $, some factors can be ignored when $ lambda $ It's big enough, so I hope there's a limit to the size of the lid volume.