Arbitrary dimensional object with constant ratio of volume to containing hyper cube's volume? Let me introduce what I need by giving an example which is not working:
n-sphere with radius 1 and respective hypercube with edges of length 2.
The ratio reaches already in 10th dimension a level of less than 1%.
My question essentially is whether a geometric object G with following features exists:


*

*The hyper-cube containing G is canonically defined for all dimensions with edges of length L and center is the 0.

*G is contained in above hyper-cube for all dimensions.

*The ratio of the G's volume and the hyper-cubes volume is constant over all dimensions.

*Calculating whether a point chosen from the hyper-cube uniformly at random is contained in G is canonical. (in other words: the calculation can be written down as a single formula depending on the number of dimensions and of course L and the point)

*Any path along the surface of G is differentiable.

*The volume ratio should be either adjustable or between .2 and .8.
(point 5 is maybe not correctly formulated. I want to make sure that G is not "edgy"/"weird" (sorry for these terms). Maybe it is the same as when I request G to be Riemann integrable)

I need such an object to generate a sample data set from an arbitrary dimensional space to benchmark performance of SVMs (and other ML algorithms) over dimensionality of input space.
 A: The following is the idea put forward by TZakrevskiy in his comment: Fix $n\geq2$,  let $C:=\{x\in{\mathbb R}^n\>|\>\|x\|_\infty\leq1$ be the hypercube of side length $2$, and let $B$ be the unit ball in ${\mathbb R}^n$. For $0<\epsilon<1$ the Minkowski sum
$$G:=(1-\epsilon) C +\epsilon B$$
is a "rounded cube" with $C^1$-surface. There exists a famous formula for ${\rm vol}(G)$, but it is obvious that
$${{\rm vol}(G)\over {\rm vol}(C)}>(1-\epsilon)^n\ ,$$
which can be made $>1-\delta$ for any given $\delta>0$.
There can be no question of "any path along the surface of $G$ being differentiable", for any $G$ proposed whatsoever. However one can say the following: Since the $G$ above has a $C^1$-surface it makes sense to consider differentiable curves on $\partial G$.
PS: I don't understand why you didn't like Will Jagy's answer, so that it was subsequently deleted. 
A: The space defined by the solutions of the following inequality seems to do the trick (d being the dimensions of the space):
$$1-(\sum_{i=1}^{d}x_i^{2d})^{1/d} \geq \frac{1}{d}$$
The space for d=2 and d=3 looks "pleasant":
 
The ratio of the solution space volume and the cube also seems to be quite consistent at 0.5:

library(rgl)

vol <- function(d) {    
    p <- function(x) 1 - sum(x^(2*d))^(1/d)
    g <- matrix(runif(d*100000, -1, 1), ncol=d)
    s <- apply(g, 1, p)
    v <- ifelse(s >= 1/d, 1, -1)

    if(d==2) plot(g[v==1,], xlim=c(-1,1), ylim=c(-1,1))
    if(d==3) plot3d(g[v==1,], xlim=c(-1,1), ylim=c(-1,1), zlim=c(-1,1))

    sum(v == 1) / nrow(g)
}

ratios <- sapply(1:100,vol)

plot(ratios, xlab="dimension")

