Solution
= Solution
Differentiate the homogeneity identity with respect to $t$ at one to obtain Euler's identity
$$
\sum_{j=1}^kx_j\partial_jp(x)=mp(x).
$$
On $X_a$, its right side is $ma\ne0$, so $dp_x\ne0$. Thus $a$ is a <regular value>. The <preimage theorem> says that the nonempty inverse image of a <regular value> of a <smooth map> $M^k\to N^l$ is a smooth submanifold of codimension $l$. Applied to $p:\mathbb R^k\to\mathbb R$, it gives \b[$\dim X_a=k-1$].